Download Ionization in atmospheric-pressure Helium plasma

Ionization in atmospheric-pressure Helium plasma jets
Pedro Arsénio Nunes Aleixo Viegas
Thesis to obtain the Master of Science Degree in
Engineering Physics
Supervisors: Prof. Vasco António Dinis Leitão Guerra
Prof. Luís Paulo da Mota Capitão Lemos Alves
Examination Committee
Chairperson: Prof. Maria Teresa Haderer de la Peña Stadler
Supervisor: Prof. Vasco António Dinis Leitão Guerra
Members of the Committee: Dr. Olivier Guaitella
Dr. Nuno Rombert Pinhão
June 2015
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Acknowledgments
Firstly, I would like to thank my supervisors Profs. Vasco Guerra and Luı́s Lemos Alves, for the opportunity of working in this subject in the IST, for their guidance in the development of this work, for the
availability to approach the problems with me, for the encouragement to keep up the good work and for
their contribution to my professional formation and to my future. I thank the whole Gas Discharges and
Gaseous Electronics (GEDG) group of IPFN for providing funding for this work and a good work environment for me. A special mention goes to my coworkers in the GEDG, Adriana Annusová and Philippe
Coche, for proficuous discussions about the development of our works and about Matlab programming,
for attending my doubts and for the sharing of problems, which has probably saved me dozens of hours
of lost endeavor. It is a pleasure to be part of the IST-LoKI team and to see this project move forward
thanks to the work of all of us.
I wish to thank everyone making possible and profitable my experience as Erasmus student in France,
which has largely contributed to this thesis. Thank you to everyone involved in the M2 “Plasmas, de
l’espace au laboratoire”, that has contributed to my formation in plasma physics and has lead me to
making a four-month internship in the EM2C (Energétique Moléculaire et Macroscopique, Combustion)
research laboratory at École Centrale Paris. I am particularly grateful to my supervisors in the EM2C,
Anne Bourdon and François Péchereau, that have guided the development of my work, that have shown
availability to solve problems together and that have decisively contributed to my formation in physics,
in programming and in teamwork and to my perspectives of future. Our meetings have been particularly
productive, either in the EM2C and with other groups of researchers, and it was really enjoyable to see
the progresses derived from our collective work. I also want to thank Deanna Lacoste for the opportunity
of acquiring competences in experimental plasma physics in the EM2C. I thank everyone in the EM2C for
making my life easier, the stay enjoyable, work more profitable and my experience richer. I also want to
show gratitude to the Laboratoire de Physique des Plasmas of École Polytechnique for providing funding
for my internship in the EM2C and to Dr. Pascal Chabert for supervising this process.
I will be forever grateful to my family for the love and support received throughout the years, in
particular to Isabel Nunes and Carlos Viegas. For giving me the opportunity to study and to move
forward during my whole life.
Thank you to all the friends and comrades showing their support throughout life and during the
development of this thesis.
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Este trabalho foi financiado por uma Bolsa de Iniciação Cientı́fica no âmbito do
projeto/instituição de I&D, Incentivo-Descargas em Gases, 4467-Inc/3
Incentivo/FIS/LA0010/2014, financiado por fundos nacionais através da portuguesa
Fundação para a Ciência e Tecnologia FCT/MEC (PIDDAC).
This work has been financially supported by a fellowship of scientific initiation under
the project/R&D institution, Incentivo-Descargas em Gases, 4467-Inc/3
Incentivo/FIS/LA0010/2014, funded by national funds through the Portuguese
Fundação para a Ciência e Tecnologia FCT/MEC (PIDDAC).
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Resumo
Esta tese de mestrado é o resultado do trabalho desenvolvido no laboratório EM2C (Énergétique Moléculaire
et Macroscopique, Combustion) na École Centrale Paris e no GEDG do IPFN no Instituto Superior
Técnico - Universidade de Lisboa. Por isso beneficia das competências de ambos os laboratórios e do
trabalho com diferentes equipas de investigação, o que providencia um contexto único de temas de investigação, modelos e ferramentas de trabalho.
As aplicações medicinais levantam desafios para a comunidade dos plasmas. São necessários plasmas
que funcionem no nosso ambiente e que se possam propagar em espaço aberto, à pressão atmosférica e
à temperatura ambiente. Jactos de plasma, onde uma descarga é guiada por um fluxo gasoso em tubos
e no ar, são apropriados para estas aplicações. O gás mais usado é o Hélio, misturado com pequenas
quantidades de Azoto e Oxigénio. A optimização das aplicações e a compreensão dos processos que
ocorrem nestes plasmas dependem da cinética das espécies no plasma.
O trabalho desenvolvido para esta tese inclui o desenvolvimento da ferramenta numérica IST-LoKI
para o estudo da cinética dos plasmas.
A cinética electrónica em misturas He-N2 -O2 é estudada,
mostrando-se os efeitos do Hélio em estado excitado, da mistura dos gases moleculares ao Hélio e da
electronegatividade do Oxigénio. São comparados modelos cinéticos para Hélio puro em estado estacionário e são obtidos resultados para diferentes condições de descarga. Finalmente, o efeito de um
campo eléctrico transitório num plasma He-N2 é simulado usando a ferramenta ZDPlasKin.
Palavras-chave:
Jactos de plasma, Hélio, Cinética electrónica, Cinética quı́mica, Ionização,
Aplicações medicinais dos plasmas
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Abstract
This master thesis is the result of work developed in both the EM2C (Énergétique Moléculaire et Macroscopique, Combustion) laboratory at École Centrale Paris and the GEDG group of IPFN at Instituto
Superior Técnico - Universidade de Lisboa. Therefore, it benefits from the competences of both laboratories and from working with different teams of researchers, which has provided a unique background of
research subjects, models and working tools.
Medical applications raise challenges for the plasma community. There is a need for plasmas that
operate in our environment and that are suitable for propagation in open space, at atmospheric pressure
and room-temperature. Plasma jets, where a discharge is guided by a gas flow through tubes and in air,
are suitable for these applications. The usual gas is Helium, mixed with small quantities of Nitrogen
and Oxygen. The optimization of applications and the understanding of the processes occurring in these
plasmas depends on the kinetics of the species in the plasma.
The work developed for this thesis includes the development of the numerical tool IST-LoKI for the
study of plasma kinetics. The electron kinetics in He-N2 -O2 mixtures is studied, depicting the effects
of excited-state Helium, of the admixture of the molecular gases to Helium and of the electronegativity
of Oxygen. Kinetic models for pure Helium in steady-state are compared and results are obtained for
different discharge conditions. Finally, the effect of a transitory electric field in a He-N2 plasma is
simulated using the tool ZDPlasKin.
Keywords:
Plasma jets, Helium, Electron kinetics, Chemical kinetics, Ionization, Medical ap-
plications of plasmas
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Contents
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iii
Resumo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix
I
Introduction and state of the art
1
I.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
I.1.1
Atmospheric-pressure non-equilibrium plasmas . . . . . . . . . . . . . . . . . . . .
1
I.1.2
Helium plasma jets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
I.1.3
Applications of Helium-based discharges: plasma medicine . . . . . . . . . . . . . .
5
I.1.4
Motivations of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
Gas discharge kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
I.2.1
Kinetic modeling of high-pressure plasmas . . . . . . . . . . . . . . . . . . . . . . .
9
I.2.2
Discharge kinetics in Helium and in Helium with admixture of air components . .
10
Background and starting point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
I.3.1
Fluid model for propagation of Helium plasma jets in tubes . . . . . . . . . . . . .
16
I.3.2
Kinetic models for Helium-based discharges . . . . . . . . . . . . . . . . . . . . . .
17
Plan of the thesis and original contributions . . . . . . . . . . . . . . . . . . . . . . . . . .
19
I.2
I.3
I.4
II Electron kinetics in Helium-based plasmas
21
II.1 General formulation for the numerical solution of the electron Boltzmann equation . . . .
21
II.2 Validation of collisional data
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
II.2.1 Solution for Helium plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
22
II.2.2 Solution for Nitrogen, Oxygen and dry air plasmas . . . . . . . . . . . . . . . . . .
25
II.3 Mixtures of ground-state Helium, Helium excited states, Nitrogen and Oxygen . . . . . .
27
II.3.1 Influence of Helium excited states on the electron kinetics . . . . . . . . . . . . . .
27
II.3.2 Influence of N2 , O2 and dry air admixture to Helium on electron kinetics . . . . .
31
xi
III Kinetic study of a pure Helium atmospheric-pressure plasma
39
III.1 Helium collisional-radiative model and validation . . . . . . . . . . . . . . . . . . . . . . .
40
III.2 Atmospheric-pressure discharge results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
IV Zero-dimensional simulation of an atmospheric-pressure He-N2 tube streamer
57
IV.1 Formulation and tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
IV.2 Reaction schemes, results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
IV.2.1 Species evolution and kinetic schemes . . . . . . . . . . . . . . . . . . . . . . . . .
59
IV.2.2 Influence of N2 concentration in the He-N2 plasma . . . . . . . . . . . . . . . . . .
68
V Conclusions and future work
75
Bibliography
79
A Electron kinetics in Helium-based plasmas
85
B Zero-dimensional simulation of an atmospheric-pressure He-N2 tube streamer
93
xii
List of Tables
III.1 Reaction scheme for He at atmospheric-pressure, reduced from [12]. . . . . . . . . . . . . .
41
B.1 Original kinetic scheme for a He plasma with 1000 ppm of N2 from [26]. . . . . . . . . . .
93
B.2 List of proposed reactions to add to those of table B.1. . . . . . . . . . . . . . . . . . . . .
94
B.3 List of proposed reactions to add to those of tables B.1 and B.2. . . . . . . . . . . . . . .
94
xiii
xiv
List of Figures
I.1
Biomedical application by propagation of jets through a thin tube . . . . . . . . . . . . .
4
I.2
Jet propagation: plasma bullet and ring-shaped electron density . . . . . . . . . . . . . .
5
I.3
Antitumor effect of plasma treatment on mice . . . . . . . . . . . . . . . . . . . . . . . . .
7
I.4
Discharge propagation velocity and structure versus N2 concentration . . . . . . . . . . .
12
I.5
Density of the most relevant species for a 99.5% He - 0.5% N2 Patm RF µdischarge . . . .
13
I.6
Electron energy distribution function for different percentages of N2 in He-N2 . . . . . . .
15
I.7
Simulation of a He-N2 discharge propagation in a tube . . . . . . . . . . . . . . . . . . . .
17
II.1 EBE-calculated and Maxwellian EEDFs for ground-state Helium . . . . . . . . . . . . . .
23
II.2 Electronic diffusion and mobility for ground-state Helium . . . . . . . . . . . . . . . . . .
24
II.3 Ionization coefficient and characteristic energy for ground-state Helium . . . . . . . . . . .
25
II.4 Reduced Townsend ionization coefficient for Nitrogen and Oxygen . . . . . . . . . . . . .
26
II.5 Reduced Townsend effective ionization coefficient for Air . . . . . . . . . . . . . . . . . . .
26
II.6 Calculated EEDFs for E/N = 1 Td for He states mixtures . . . . . . . . . . . . . . . . .
28
II.7 Calculated EEDFs for E/N = 10 Td and E/N = 50 Td for He states mixtures . . . . . .
29
II.8 Ionization coefficient in mixtures of He(11 S) and He(23 S) . . . . . . . . . . . . . . . . . .
30
II.9 Calculated EEDFs for E/N = 10 Td for several He-N2 and He-O2 mixtures . . . . . . . .
32
II.10 Calculated EEDFs for E/N = 10 Td for He mixtures with dry air and for E/N = 1 Td
for excited He mixtures with O2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
II.11 Ionization coefficient in He-N2 mixtures as function of E/N and densities . . . . . . . . .
33
II.12 Ionization coefficient in He-O2 mixtures as function of E/N and densities . . . . . . . . .
34
II.13 Ionization coefficient in He-air mixtures as function of E/N . . . . . . . . . . . . . . . . .
35
II.14 Ionization coefficient and power transfered to ionization in He-air mixtures as function of
the relative densities of air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
III.1 Reduced electric fields and He ground-state densities as f(ne ) from [12] and from IST-LoKI
using the reaction scheme from table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
III.2 He ions relative densities as f(ne ) from [12] and from IST-LoKI using the reaction scheme
from table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
44
III.3 Relative creation rates of electrons and ions through associative ionization and stepwise
ionization as f(ne ) from [12] and from IST-LoKI using the reaction scheme from table III.1 45
III.4 Relative loss rates of electrons and ions through dissociative recombination and diffusion
of charged species as f(ne ) from [12] and from IST-LoKI using the reaction scheme from
table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
III.5 He(23 S) and He(21 S) relative densities as f(ne ) from [12] and from IST-LoKI using the
reaction scheme from table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
III.6 He(23 P ) and He∗2 relative densities as f(ne ) from [12] and from IST-LoKI using the reaction
scheme from table III.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
III.7 E/N (Td) and Te (eV) as f(ne ) from IST-LoKI using the reaction scheme from table III.1
49
III.8 Temporal evolution of the He species densities from IST-LoKI using the reaction scheme
from table III.1 and ne = 1013 cm−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
III.9 Charged species and excited species densities as f(ne ) from IST-LoKI using the reaction
scheme from table III.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
III.10Rates of creation and destruction of He+
2 as f(ne ) from IST-LoKI using the reaction scheme
from table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
III.11Rates of creation and destruction of He+ as f(ne ) from IST-LoKI using the reaction scheme
from table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
III.12Rates of creation and destruction of electrons as f(ne ) from IST-LoKI using the reaction
scheme from table III.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
III.13Rates of creation and destruction of He(23 S) as f(ne ) from IST-LoKI using the reaction
scheme from table III.1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
III.14Rates of creation and destruction of He∗2 as f(ne ) from IST-LoKI using the reaction scheme
from table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
IV.1 Densities in the He plasma with 1000 ppm of N2 using the reaction scheme of table B.1 .
60
IV.2 Rates of production of e− and of
N+
2
in the He plasma with 1000 ppm of N2 using the
reaction scheme of table B.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
IV.3 Densities of the chemically relevant species in the He plasma with 1000 ppm of N2 using
the reaction scheme of table B.1 and the 2D model of section I.3.1 . . . . . . . . . . . . .
62
+
2
IV.4 Rates of production of N+
2 (B Σu ) and rates of N2 (C → B) and N2 (B → X) in the 99.9%
He - 0.1% N2 plasma using the reaction scheme from tables B.1 and B.2 . . . . . . . . . .
63
IV.5 Rates of production of N+
2 and of the chemically relevant species in the 99.9% He - 0.1%
N2 plasma using the reaction scheme from tables B.1 and B.2 . . . . . . . . . . . . . . . .
64
IV.6 Densities of the chemically relevant species using the reaction scheme from tables B.1, B.2
and B.3, in the 99.9% He - 0.1% N2 plasma . . . . . . . . . . . . . . . . . . . . . . . . . .
65
IV.7 Densities of the chemically relevant species using the reaction scheme from tables B.1 and
B.2 without Penning ionization reactions, in the 99.9% He - 0.1% N2 plasma . . . . . . .
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67
IV.8 2D simulation results for the propagation of the 99.9% He - 0.1% N2 plasma, using the
reaction scheme from tables B.1 and B.2 without Penning reactions . . . . . . . . . . . . .
68
IV.9 Densities of the most chemically relevant species for the cases of 10 and 10 000 ppm of N2
in the He-N2 plasma, using the reaction scheme from tables B.1 and B.2 . . . . . . . . . .
69
IV.102D distribution of the N2 (C 3 Πu ) densities for 10, 100, 1000 and 10 000 ppm of N2 in the
He-N2 plasma, using the reaction scheme from tables B.1 and B.2 . . . . . . . . . . . . . .
70
IV.11Comparisons for several N2 densities in the He-N2 plasma of the temporal evolution of
the electron densities and of the creation rates by Penning ionization, using the reaction
scheme from tables B.1 and B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
71
IV.12Electron density in different He-N2 mixtures, using the reaction scheme from tables B.1
and B.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
IV.13Electron density in different the 99.9% He - 0.1% N2 plasma with different pre-ionization
values, using the reaction scheme from tables B.1 and B.2 . . . . . . . . . . . . . . . . . .
73
A.1 Interpolation of electron-He(11 S) collision cross-sections . . . . . . . . . . . . . . . . . . .
85
A.2 Calculated and Maxwellian EEDFs for ground-state Helium (1 Td, 10 Td, 50 Td and 100
Td)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
A.3 Calculated and Maxwellian EEDFs for ground-state Helium (250 Td, 500 Td and 1000 Td) 86
A.4 Swarm parameters for Nitrogen and Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . .
87
A.5 Reduced mobility and characteristic energy for Air . . . . . . . . . . . . . . . . . . . . . .
88
A.6 Most relevant electron-He collision cross-sections . . . . . . . . . . . . . . . . . . . . . . .
88
A.7 Calculated EEDFs for E/N =1 Td for He states mixtures . . . . . . . . . . . . . . . . . .
89
1
3
A.8 Swarm parameters for different mixtures of He(1 S) and He(2 S) . . . . . . . . . . . . . .
89
A.9 Swarm parameters for different mixtures of He(11 S), He(23 S) and He(21 S) . . . . . . . .
90
A.10 Calculated EEDFs for E/N =1 Td and E/N =50 Td for He-N2 mixtures . . . . . . . . . .
91
A.11 Calculated EEDFs for E/N =1 Td and E/N =50 Td for He-O2 mixtures . . . . . . . . . .
91
A.12 Calculated EEDFs for E/N =1 Td and E/N =50 Td for He-air mixtures . . . . . . . . . .
92
A.13 Calculated EEDFs for E/N =10 Td and E/N =50 Td for He-O2 mixtures, including He
excited states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
B.1 Rates of some chemically relevant reactions using the reaction scheme from tables B.1, B.2
and B.3, in the 99.9% He - 0.1% N2 plasma . . . . . . . . . . . . . . . . . . . . . . . . . .
95
B.2 Densities of the chemically relevant species using the reaction scheme from tables B.1 and
B.2 without charge transfer reactions, in the 99.9% He - 0.1% N2 plasma . . . . . . . . . .
95
B.3 Densities of the chemically relevant species using the reaction scheme from tables B.1 and
B.2 without 3-body reactions, in the 99.9% He - 0.1% N2 plasma . . . . . . . . . . . . . .
96
B.4 Densities of the most chemically relevant species for the cases of 100 and 30 000 ppm of
N2 in the He-N2 plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
B.5 Axial evolution of the e− densities for the cases of 10, 100, 1000 and 10 000 ppm of N2 in
the He-N2 discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
97
B.6 Axial evolution of the longitudinal E field for the cases of 10, 100, 1000 and 10 000 ppm
of N2 in the He-N2 discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xviii
97
Nomenclature
Abbreviation
Definition
E
Electric field
E/N
Reduced electric field
e−
Electron or electrons
EBE
Electron Boltzmann Equation
EEDF
Electron Energy Distribution Function
fe
Electron Energy Distribution Function
He
Atomic Helium
N
Gas density
N
Atomic Nitrogen
N2
Molecular Nitrogen
ne
Electronic density
O
Atomic Oxygen
O2
Molecular Oxygen
p
Pressure
P
Power
Patm
Atmospheric-pressure
T
Temperature
Tgas
Gas Temperature
X+
Positive ion of species X
X
−
Negative ion of species X
xix
xx
Chapter
I
Introduction and state of the art
I.1
I.1.1
Introduction
Atmospheric-pressure non-equilibrium plasmas
Cold plasmas are weakly ionized gases where ions, electrons and neutral species coexist, although the
electron density ne is much lower than the neutral density n0 (ne /(ne + n0 ) < 0.1 [1]). This means that
electromagnetic phenomena exist inside the plasma due to the charged species, even though a condition
of global quasi-neutrality (equal density of negative and positive charges n− ' n+ ) is followed. These
species are not in thermal equilibrium with each other and usually heavy species (ions and neutrals)
stay near room temperature (300 K, Tgas < 104 K) while electrons are accelerated to kinetic energies
in the 1-30 eV range (104 − 105 K). Electrons are therefore just a few but they are very energetic,
while the most part of the gas volume is cold and non-damaging to the surrounding environment. Ions
and neutral species exist not only in their ground-states but also in excited rotational, vibrational and
electronic levels which usually are not in thermal equilibrium among themselves and which carry internal
energy. This non-equilibrium leads to a high-level of reactivity inside the plasma, since species carry
kinetic and internal energy which can be exchanged by means of collisions, and this will be determinant
for the development of the plasma. Therefore, cold plasmas are described as very chemically reactive,
energetically economic (no or moderate heating) and non-destructive. These characteristics raise interest
in cold plasmas not only from a fundamental point of view but also due to their potential applications,
especially if it is possible to control the creation of plasma species. Laboratory plasmas are characterized
by the gas they are created from, the way they are created and sustained, the geometry in which they
exist and the gas density used to produce them. Non-equilibrium plasmas are often created by applied
electric fields with different shapes in space and in time [2].
During the 20th century, plasmas have been extensively studied at low-pressure (p ∼ mTorr) or
equivalently at low gas density (n0 = P/(KB T0 ) = 1012 − 1014 cm−3 ) for use in various applications. In
this case there is a higher mean free path of electrons (hλi = (n0 hσe0 i)−1 ) and a lower electron-neutral
collision frequency (νe0 = n0 hσe0 (ve )ve i) than at higher pressures and the plasma processes occur slower,
which allows a better understanding and control of the plasma properties [2].
1
In the last thirty years, there has been an increasing interest about plasmas at higher pressures
(p ∼ 0.1 − 10 bar, 100s Torr, 104 − 106 Pa) and particularly about plasmas at room temperature and
atmospheric-pressure (n0 = 2.45 × 1019 cm−3 ), since in some cases they are not confined to vacuum
chambers, thus avoiding expensive pumping systems. In addition, they are more compatible with the
environment we live in [1].
At high pressures, the physical mechanisms leading to the development of the plasma are very different
from those at low pressures. During the application of an electric field to a gas, as the frequency of
electron-neutral collisions is high, electrons have less time to acquire enough kinetic energy from the
applied field to provoke ionizing collisions. Therefore, the ionization efficacy by electron-neutral collisions
is lower than at low pressures. This means that, even though the number of collisions is higher at
higher pressures, the applied electric field to create a plasma is usually higher than at low/intermediate
pressures, around 10s kV/cm [3, 4]. In air, for instance, the applied field between electrodes should be
around 30 kV/cm for the discharge breakdown, which implies an inter-electrode space in the cm or mm
order of magnitude. The size of objects to be treated by these plasmas is therefore limited. This is a
problem in particular for biomedical applications, in which case the objects to be treated usually have
bigger dimensions [5].
In reality, this electric field is not applied equally to the whole volume of gas and ionization is
strongly localized leading to very heterogeneous plasmas. The atmospheric-pressure plasma often has a
filamentary structure and propagates as a channel with diameter ∼ 100s µm, in which case it is called
a micro-discharge and there is a big concentration of electrons. This type of discharge has a strong
concentration of active species and local heating may be non-negligible. The natural tendency towards a
filamentary structure, ensuring non-homogeneity, is a problem for applications such as surface treatment
or deposition, where the use of low-pressure plasmas is preferable [1, 3]. In order to avoid gas heating,
the electron density has to remain weak. One way of avoiding the transition towards a thermal arc is the
insertion of dielectric barriers between electrodes, which limits the current flowing through the plasma.
In this case, the plasma is said to be produced by a Dielectric Barrier Discharge (DBD). Another way of
avoiding the arc thermalization is the application of a pulsed voltage with short time-lengths, typically
under hundreds of nanoseconds, in which case the plasma is called a nanosecond pulsed discharge [4].
Atmospheric-pressure plasmas can be described by the same set of fluid equations as low-pressure
collisional plasmas. These are the particle continuity equation, the drift-diffusion momentum conservation
equation and the Poisson’s equation, respectively [3]:
∂ns
+ ∇.(ns v~s ) = Ss − Ls
∂t
~ − Ds ∇ns
j~s = ns v~s = ns µs E
P
~ = i qi ni
∇.E

 X
Y

Ss , Ls =
ni  Kj
j
i=1,2,3
2
(I.1)
(I.2)
(I.3)
(I.4)
Ds =
kB Ts
qs
, µs =
ms νs0
ms νs0
Kj , Ds , µs = f (E/N, gas mix)
(I.5)
(I.6)
In these equations, ns is the number density of species s, v~s is its velocity and j~s its flux. Ss and
Ls are, respectively, the rates of production and loss of the species s by collisional-radiative processes.
For the case of kinetic processes, they are given by the densities of 1, 2 or 3 species ni that take part
in each reaction, multiplied by the coefficients of reactions j, Kj . µs is the electric mobility and Ds is
the free diffusion coefficient, with kB the Boltzmann constant, Ts the temperature of species s, ms its
~ = −∇V , where
mass, qs its charge and νs0 its electron-neutral collision frequency. The electric field is E
V is the electric potential and is the dielectric permittivity of the medium. The reduced electric field
is the ratio between the electric field and the gas density E/N . The electric field governs the energy
gained by electrons between two successive collisions and the gas density controls the mean free path
and collision frequency of electrons. This leads to electron energy distribution and discharge properties
typically dependent of E/N that makes it a scaling parameter.
Unlike low-pressure plasmas, the high local heterogeneity in high-pressure discharges means that
charges in the plasma create relevant electric fields. Therefore, the transport and reaction rate coefficients
(detailed in section I.2.1) no longer depend only on the electric field imposed externally but also on
the electric field created locally, obtained from Poisson’s equation (I.3). The assumption of a thermal
equilibrium with the local field means that the discharge propagation depends on the local distribution
of charges and that space-charge mechanisms govern the dynamics of the discharge, which explains the
mentioned filamentary structure [1]. In fact, space-charge mechanisms are not enough to understand
the development of high-pressure discharges on very short time scales (ns). In 1939-1941, Meek, Loeb
and Raether introduced the concept of ionization wave that leads to the streamer model, to characterize
filamentary type inhomogeneous non-stationary discharges. This model will not be the object of further
development in this work but its knowledge is required to explain the development of plasmas in the next
sections.
The possibility of propagating cold plasmas at atmospheric-pressure (Patm ) into open space is very
interesting. Such systems are called Atmospheric-Pressure Non-Equilibrium Plasma-Jets (APNP-Js) or
just Jets, and will be further described in section I.1.2. They can propagate away from the electrodes
into open space and touch the human body directly. This propagation can be made inside dielectric
tubes, allowing the plasma to be applied somewhere farther (10s cm) without spatial confinement, with
no limitation on the size of the object to be treated. The difference between the propagation in a tube and
in open space is due to the surface charges on the tube walls that influence the electric field distribution.
Due to emerging applications, APNP-Js have attracted a lot of attention in the past decade. One example
of jets with very practical application can be seen in figure I.1.
Industrial and technological usage of atmospheric-pressure plasmas includes elemental analysis, thinfilm deposition and etching, environmental applications, such as treatment of gas pollutants (e.g. ozone
in water treatment), production of photo-voltaic cells, air depollution and cleaning, low consumption
lighting and energy production (e.g. from the reforming of gases), in addition to biomedical applications,
3
Figure I.1: Example of a biomedical application by propagation of jets through a thin tube [6].
such as sterilization, skin regeneration and cancer treatment [7–11], that will be further detailed in
section I.1.3. Floating electrode DBDs, atmospheric-pressure torch and plasma jets in continuous, radiofrequency or pulsed regimes are the main discharge setups used for the production of non-thermal rare
gas plasma expanding in ambient air [11, 12]. These setups can be safe, reliable, compact and easy to
handle, but they may suffer from two potential limitations, when envisaging medicine protocols. On one
hand, plasma torches and jets allow for surface treatment areas with typical diameters of only a few
hundreds of microns. On the other hand, DBD systems, although operating efficiently on larger surfaces,
require a gap between the high voltage reactor electrode and the tissue of only a few millimeters [11]. As
far as APNP-Js are concerned, they have led to many cutting-edge applications in medicine, health care,
food processing and nanotechnology, since these atmospheric-pressure plasmas can interact with living
tissues, cells and bacteria without causing thermal damage.
I.1.2
Helium plasma jets
Following the previous section, the development of APNP-Js raises interest. Different gases have been
studied for jet propagation but usually the jets work with noble gases mixed with a small percentage of
reactive gases [13]. It is difficult to generate atmospheric-pressure (Patm ) N2 plasma jets due to the low
energy excitation reactions in N2 that make electron-impact ionization harder. Also, due to the presence
of 21% electronegative O2 and of the consequent attachment reactions, it is difficult to sustain APNP-Js
in air. The Helium jet is the most studied case and the one that has shown longer propagation lengths in
air [5, 13]. This happens due to the high direct electron-impact ionization coefficient for He at the typical
reduced electric fields that create jets, below 200 Td (1 Td = 10−21 V.m2 ), which makes propagating a
jet in He much easier than in air, that has a lower ionization coefficient. A He plasma jet is therefore a
very efficient way to generate and control non-equilibrium plasmas at Patm , even at low power densities,
thus limiting the development of thermal instabilities [14].
It has been observed that the plasma jet propagates along the He flux in open space like an ionization
wave, which means that the flow of He guides the discharge from the tube exit, that serves as anode,
4
towards the air, that serves as cathode. Confining the plasma in the flow prevents also its branching and
radial expansion. The He flow has to be in laminar mode at the output of the dielectric tube to produce
a plasma jet. The big difference between the propagation of a classic streamer and that of a jet is that
the jet is confined by the gas flux and the tube geometry and is guided by that flux far away from the
electrodes, whereas the streamer propagates between two electrodes [14]. Figure I.2 shows observations
of the propagation of an APNP-J and the simulation of the electron density in it.
Figure I.2: Jet propagation: the observed luminosity in the shape of a plasma bullet [15] and the simulated
ring-shaped electron density [14].
A potential drop along the quasi-neutral channel between the anode and the discharge front increases
linearly with length, thanks to the increase in the electrical resistance. This means that the propagation
length in open air depends on the potential of the discharge front at the tube exit and on the conductivity
of the channel. Therefore, long life-time species and active plasma chemistry in the channel can affect the
plume propagation. Globally, the propagation velocity and the length of the jet depend mostly on the
voltage at breakdown, the electrode geometry, the working gas, the shape of the applied voltage (notably
of its duration) and the gas flow [5, 13–15].
Moreover, the mentioned reactive species may play an important role in applications, as it will be seen
in section I.1.3. In fact, the transient nature and the associated thermal non-equilibrium of the guided
ionization waves of APNP-Js enable many interesting physical and chemical effects, such as highly-reactive
chemistry at near-room temperature, effective and fast transport of charged species to the surfaces being
processed and controllability of various agents, like radicals, ions, UV radiation and electric fields [5, 7].
I.1.3
Applications of Helium-based discharges: plasma medicine
Plasma medicine has been mentioned as a field of application of Patm plasmas and of APNP-Js. It
is an emerging field leading to multidisciplinary research works attracting researchers from different
disciplines such as engineering, plasma physics, chemistry, biology and medicine. Promising applications
in medicine involve electro-surgery, dentistry, skin care, treatment of mammalian and cancerous cells,
blood coagulation, wound healing and sterilization of heat-sensitive medical instruments. This new
field presents technological challenges for developing plasma sources and raises fundamental questions
regarding physical phenomena. Mechanisms of plasma interaction with cells and living tissues are still
unclear due to the complexity of both plasma and biological systems. Since it is difficult to investigate
the characteristics of all the agents involved in these processes, plasma simulation and modeling have
been widely adopted in an attempt to understand each phenomenon, the coupling between phenomena
and the underlying physics [7].
5
Noble gases such as He and Ar are widely used to generate plasmas for medical applications, since they
result in the stable generation of atmospheric-pressure discharges at low gas temperature, and because
they can lead to the production of energetic species when converted into plasma jets interacting with the
N2 and O2 in open air. Molecular gases such as O2 are often added to the noble gases to enhance the
generation of reactive radicals and ions, through collisions between electrons and neutrals, while the gas
remains nearly at room temperature [7, 12].
It has been suggested replacing light sources by plasma jets, in combination with hydrogen peroxide
(H2 O2 ), for teeth bleaching in dental clinics [7]. It was demonstrated that a tooth exposed to a He jet
with H2 O2 becomes brighter when compared with a tooth treated with H2 O2 alone. Acceleration of tooth
bleaching by the jet results from the enhancement of OH radical production due to the H2 O2 exposure
to the plasma. For instance, the jet was effective in bleaching teeth stained by wine or coffee [7].
For caries and root canal treatment there is also the need for new methods that can completely sterilize
the infected dental tissues. In [8], it was demonstrated that an APNP-J of He is effective in sterilizing
three different kinds of oral pathogenic microorganisms. For planktonic microorganisms in suspension,
the sterilization effect is pH dependent and is proeminent below a pH of 4.5. The effect is due to Reactive
Oxygen Species (ROS) (O−
2 and HO2 ).
Traditional techniques for bacteria inactivation, like autoclaving and ethylene oxide treatment, can be
hazardous, time consuming and not suitable for heat sensitive materials or food packaging. Patm plasmas
provide an alternative method that is safe, convenient and fast to operate for microbial inactivation,
which eliminates harmful gas emissions and that makes operation possible at high rates with reduced
temperature. Sterilization experiments have been conveniently performed in an open-air environment,
with an exposure time to a jet of only 1 s, achieving five decades of inactivation [9]. The plasma plume had
a lethal effect on all tested microbes. Also, the afterglow plume portion of the plasma was implemented
for microbial inactivation, which differs significantly from using the active discharge. The afterglow plume
temperature is less than 50o C, which is especially valuable for treating heat-sensitive samples. In the
afterglow region there are fewer charged particles due to the recombination processes within it, which
also contributes to reduce the damage of the workpiece surface by the positive ions accelerated in the
floating sheath [9].
APNP-Js generated in He by high-voltage pulses were successfully used in the treatment of skin burnwounds of rats in [10]. Many excited and reactive species, such as ROS and Reactive Nitrogen Species
(RNS), including OH, O and NO, act as strong oxidizing agents. In particular, the highly reactive OH
radical can intensify the oxidative stress on cells. These radicals are critical to cell health and their
presence makes the jet suitable for this application when a proper He/O2 mixture ratio is used.
In [16] it was found that the use of a He plasma jet with a small admixture of O2 (∼ 2%) can
reduce the rates of the HIV-1 infection of macrophages. Macrophages, cells of innate immune system,
constantly circulate for tissue surveillance. The impact of the jet on these cells was evaluated through
viability studies to estimate the rate of induced cell infection. It was found that the jet does not affect
the viability of the human macrophages, although it can reduce the rate of the HIV-1 infection in the
infected macrophages.
6
Cancer cell treatment has been extensively reported in [11]. In Europe, the estimated new cancer cases
in 2006 were about 3.2 million, representing 1.7 million deaths. In some tumor types, the therapy outcome
remains poor, especially when recurrence occurs. To increase the cancer-patient survival, improvement
of current therapies and new therapeutic concepts are needed. Current research efforts are focused on
the development of cancer specific therapies with little or no toxic side effects, such as local anti-tumor
treatment with APNP. Plasma, as an active ionized medium sustained by different forms of energy
(electric, thermal and radiative), can induce results similar to those usually obtained through chemical
or radiotherapy treatments. It has been demonstrated that plasma treatment induces cell death in a
melanoma cell line. Mechanisms underlying this apoptotic process in these cancer cell lines remain rather
unclear and need further investigation. The work in [11] was conducted to evaluate the potential antitumor effect of an in vivo plasma treatment, on a U87-luc glioma tumor. A ms pulsed DBD at moderate
power was used for treatment, using for example a He-air mixture. The plasma treatment was shown to
be safe for mice, even if the long exposure time (20 min, 3 consecutive days) produces a superficial burn.
After 5 days of plasma treatment, a dramatic U87 bio-luminescence decrease was observed, associated
with a reduction in the tumor volume of the U87 glioma-bearing mice, as shown in figure I.3 [11].
Figure I.3: Antitumor effect of plasma treatment. The mouse on the right was not treated and the mouse
on the left was irradiated by plasma, for 5 consecutive days at 100 Hz during 6 min [11].
Further studies are needed to enhance the chemical activities of plasmas for medicine, so that treatment times can be further shortened and screened for safety. Also, the development of APNP-Js suitable
for large-area applications is a topic of ongoing and future research effort [13].
I.1.4
Motivations of this work
The applications presented in section I.1.3 raise challenges for the plasma community. There is a need
for plasmas that operate in our environment and that are suitable for propagation in open space, which
implies atmospheric-pressure and room-temperature. It is also required that the plasma is safe and nondestructive for biological applications, as well as energetically economic. We have seen that one example
of such plasmas are jets, in which case the discharge is guided by a gas flow through tubes and into open
space and where low to intermediate electric fields, below 200 Td, should be considered.
7
Helium is the most studied gas for such plasmas and the one that has shown longer propagation lengths.
However, in our environment, we usually need to consider the plasma interaction with air molecules. Not
only air impurities always exist at atmospheric-pressure, but propagation in open-space also implies the
variation of air densities interacting with the plasma. Besides, the species created within the surrounding
air can be used to optimize the actual application of the plasma by the action of radicals, ions, radiative
transitions or the actual electric field. It has been noted that reactive species such as O2 are often added
to Helium for application purposes and that the plasma can provide reactive oxygen nitrogen species
(RONS). Moreover, medical applications imply the interaction of the plasma with biological tissues and
demand its development and propagation adapted to medical devices, such as catheters.
Therefore, it is of paramount importance to understand the highly-reactive chemistry of plasmas at
near-room temperature, getting into detail on the generation of energetic and reactive species, in order
to control these agents that allow many of the interesting physical and chemical effects occurring in jets.
This is the way to understand the mechanisms of plasma interaction with cells and living tissues and
enhance the activity of plasmas for medicine. There is also the need to understand the development
of plasma jets and how charge production in the channel influences jet propagation. Charges in the
plasma control the electric field distribution, according to Poisson’s equation (I.3). They influence the
conductivity in the jet channel, defining the potential in the discharge front and critically affecting the
jet propagation velocity. The search for understanding on these questions leads to the study of ionization
mechanisms in the plasma.
The fact that the propagation of jets varies in He-air mixtures with the gas concentrations increases
the interest of studying the changes in the jet dynamics and structure with the composition of the
plasma forming gas [5, 14]. Given the constitution of air (78% N2 and 21% O2 ), research could start
by understanding the discharge dynamics in pure He, He with N2 impurities, He with O2 impurities
and He with dry-air (80%N2 -20%O2 ) impurities. Such a study should be done in tubes where the gas
constitution is more controlled, and which have particular interest for medicine as catheters and should
try to understand and differentiate the dominant processes in both discharge and post-discharge regions.
A deep knowledge of the fundamental processes governing the plasma medium is of great importance.
The main kinetic processes associated with each specific mixture need to be identified. In this context,
the development of state-of-the-art collisional–radiative models (CRMs) is of great interest. CRMs are
simulation tools for the kinetic description of discharge plasmas, which aim at obtaining the populations
of the different charged and excited species within the gas/plasma system as well as their effective
creation/loss rates, relating them to the discharge maintenance characteristics through the equation
system (I.1-I.6). Species temporal evolutions in the plasma are described by the particle-balance equations
(I.1) where source and loss terms (I.4) for reactions are included. In the case of jets, the accumulation of
active species, produced by a sequence of discharges, should be accounted for, in particular for the case
of species with life-times higher than the discharge repetition period [5, 14]. The results of CRMs can
be subsequently used to ‘reduce’ complex kinetic schemes and allow their utilization in heavier codes, by
revealing the dominant factors governing the different parameters and by identifying the most important
populations and rates for the atomic/molecular system under study [12].
8
I.2
I.2.1
Gas discharge kinetics
Kinetic modeling of high-pressure plasmas
CRMs require knowledge about collisional–radiative data (cross sections for interactions with electrons,
collision rate coefficients and radiative decay frequencies for interactions between heavy species) and
transport parameters, to solve the rate balance equations (I.1) of the different species in the plasma, coupled with the electron Boltzmann equation (EBE). The typical validation of a CRM involves comparisons
between predicted and measured values of the population densities with the main excited species of the
plasma, obtained under the same work conditions. Usually, the densities of excited species are measured
from optical emission spectroscopy (OES) diagnostics, also used to obtain the electron density and the
gas temperature [12].
Local properties of a filamentary plasma, such as electron density ne , electron temperature Te , E/N
and electron energy distribution function (EEDF) fe () may be hard to obtain experimentally and simulations may play an important role to fill the gaps. While ions and neutrals follow the fluid equations
presented earlier, electrons can be described by the EBE [17]:
∂fe (~r, v~e , t)
~r fe (~r, v~e , t) − e E(~
~ r, t).∇
~v fe (~r, v~e , t) =
+ v~e .∇
∂t
me
∂fe
∂t
(~r, v~e , t)
(I.7)
coll
The solution to the Boltzmann equation for a specific gas or gas mixture yields the non-equilibrium
EEDF fe , allowing to determine the electron mean kinetic energy (associated with Te ) and all the electron
transport parameters and rate coefficients, as a function of the reduced electric field E/N , the quantity
that links these parameters to the discharge equations (I.1-I.6). For the particular case of an electronimpact reaction, described by cross section σ(ve ), the rate coefficient ks writes:
Z
ks (E/N ) = hσ(ve )ve i = σ(ve )ve fe (ve , E/N )4πve2 dve
(I.8)
For collisions between neutral species, the rate coefficients usually follow the Arrhénius expression:
Eas
αs
ks (T0 ) = Bs T0 exp −
,
(I.9)
kB T0
where Eas is the activation energy per molecule, kB T0 is the average kinetic energy of the neutral species,
Bs T0αs is the total frequency of collisions and e−Eas /(kB T0 ) is the probability that any given collision will
result in a reaction.
To know the main mechanisms by which electrons dissipate energy, we must look for the dominant
electron-neutral collision cross sections at values close to the electron mean energy, which depends on
the applied E/N . Thus, controlling E/N allows guiding the creation of reactive species by electronimpact that can be useful for applications or simply for discharge maintenance. The reduced electric field
accelerates the electrons, which acquire enough energy to undergo not only elastic but specially inelastic
collisions with the gas neutral species, of which electron-impact ionization reactions are essential to firstly
create the plasma.
Besides electron-impact ionization, other important electron production processes occurring in high9
pressure discharges are photo-ionization and associative and Penning ionization processes, in which excited atoms or molecules combine or collide to form ions and free electrons. Moreover, photo-ionization
and associative and Penning ionization depend on the production of radiative and metastable excited
species, respectively, which again are mainly produced by electron-impact at high pressure. For these
conditions, both ions and electrons are lost essentially by dissociative recombination (e− + AB+ → A +
B). Attachment is another major process of electron loss in electronegative gases, like O2 (e− + O2 + M
→ O−
2 + M) [3, 18].
Electron collisions under the effect of a localized electric field take place in the nanosecond range.
Additionally, at high pressures, electron losses due to transport can be neglected, meaning that the EBE
can be simplified as to describe a local balance between the energy gained from the electric field and lost in
collisions. Chemical reactions involving neutrals, ions and electrons, such as attachment, charge transfer,
recombination and association, as well as radial diffusion of heavy species occur in the microsecond range.
Finally, reactions between neutrals in the plasma volume happen on a millisecond range and, therefore,
this is the time-scale to cover to describe the global dynamics of high-pressure plasmas [3].
I.2.2
Discharge kinetics in Helium and in Helium with admixture of air components
In the context of He applications (subsection I.1.3), the starting point must be the development of stateof-the-art CRMs for pure He at atmospheric pressure, before evolving to mixtures of He with molecular
gases. The kinetic schemes used for the evaluation of the source and loss terms in He-air mixtures should
include electron-impact excitation and ionization of atomic and molecular species, associative ionization,
Penning ionization of air molecules in collisions with excited He atoms and excimer molecules, electron-ion
recombination and attachment of electrons to oxygen molecules [5].
Discharge kinetics in pure Helium
In Santos et al. (2014) [12], an atmospheric-pressure CRM was developed, adopting a consistent set
of electron cross sections and kinetic mechanisms. The model was implemented through a numerical
tool similar to the one described in section I.3.2 (IST-LoKI [19]) and was validated by OES diagnostics
(measuring the electron density, the gas temperature and the populations of various excited states), for a
surface-wave discharge of field oscillation frequency f =2.45 GHz, produced in a tube of 3 mm inner-radius,
at Tg =1700 K and ne = 2.45 × 1013 cm−3 .
The model in [12] adopts an updated kinetic scheme that considers several electron and heavy species
collision mechanisms involving electrons e− , He(n, l, s)≡He(n2s+1 l) excited states (where n, l and s are
the principal, the orbital and the spin quantum numbers, respectively), ground-state atomic ions He+ ,
3
+
∗
ground-state molecular ions He+
2 (2 SΣu ) and He2 excimers. The CRM considers all the neutral atomic
states up to level n = 7. The kinetic scheme includes electron-impact excitation, de-excitation, direct
ionization and stepwise ionization with cross sections for electron energies up to 1 keV; recombination
and electron-impact dissociation; associative ionization mechanisms; charge transfer reactions between
10
He+ and He+
2 (including 3-body ion conversion, particularly important at high-pressures); internal energy
transfer between excited states of He and He2 ; radiative transitions between He excited states; transport
of heavy particles (He(23 S), He(21 S), He∗2 , He+ , He+
2 ), which have long lifetimes τ (e. g., the metastable
states He(23 S) and He(21 S) have τ ∼ 8400 s and 2 × 10−2 s, respectively, while the radiative states
He(23 P ) and He(21 P ) have τ ∼ 10−8 s and 10−7 s, respectively). References [20–22] present similar pure
Helium schemes. Wang et al. [22] emphasize, in particular, the need for including stepwise ionization of
−
+
−
excimers (He∗2 + e− → He+
2 + 2 e ) and the atomic ion 3-body recombination (He + He + e → He +
He(23 S)). The coefficients of heavy particle reactions are either constant or gas temperature dependent.
The metastable states He(23 S) and He(21 S) and the radiative states He(23 P ) and He(21 P ), with densities
around 1012 cm−3 , are particularly important for Penning reactions producing He+ and He+
2.
The results from [12] show that at Patm the power absorbed from the field by electrons is mostly
lost in elastic collisions with ground-state atoms, instead of inelastic collisions. Results also show that
−
(i) electrons are mainly created by associative ionization (He∗ + He → He+
2 + e ) and destroyed by
−
∗
+
dissociative recombination (He+
2 + e → He + He); (ii) He ions are mainly destroyed and created by
the three-body ion-conversion direct and reverse reactions (He+ + 2 He ↔ He+
2 + He), respectively, and
are also created by electron-impact direct ionization; (iii) in the case of low ne , He+
2 ions are created and
destroyed by associative ionization and dissociative recombination, respectively; for ne ≥ 4 × 1013 cm−3 ,
He+
2 ions are created and destroyed mainly by the three-body ion-conversion direct and reverse reactions.
Discharge kinetics in Helium with admixture of air components
Noble gas jets are relatively easy to generate, but they are not very reactive. This is why, besides air
impurities in small quantities, reactive gases are usually added to the working noble gas when jets are used
for applications. For biomedical applications, O2 or H2 O2 are usually added. For etching applications,
CF4 or O2 can be used [13].
A plasma jet of He with a small admixture of N2 is considered as a typical electropositive plasma. In
[5] it is stated that electron-impact ionization of He atoms and N2 molecules dominates in the discharge
front of APNP-Js. However, Penning ionization of N2 molecules sustained by metastable states of the
working gas is the main ionization mechanism within the body of the streamer behind the ionization
front and, therefore, controls the conductivity in the channel that sustains the jet propagation.
Experimental results on the propagation of Helium jets with Nitrogen admixture in a 40 cm long
dielectric quartz tube with a 4 mm inner diameter have been published by Darny et al. (2014) [23].
In this experiment, the N2 concentration is changed for constant operating parameters (Patm , applied
voltage, gas flow rate), which leads to the identification of the relative importance of kinetic reactions that
depend on the N2 concentration. The average velocity of propagation in the tube versus N2 concentration
is shown in figure I.4. It is shown that with just 0.1% of N2 the velocity significantly increases compared
with He with minimal impurities. The velocity reaches a maximum value around 0.25% N2 . For 0.9% N2 ,
the velocity is close to that obtained for pure Helium and it is clearly lower for higher N2 percentages. In
the same figure we can observe the structure of the discharge ionization front inside the capillary tube.
These results show how relevant the N2 variations are for the propagation dynamics and the structure
11
Figure I.4: Discharge’s propagation velocity and ionization front structure versus N2 concentration [23].
of the He jet, but they do not provide a full understanding of the phenomena. Moreover, they show how
important the modeling of discharges and of their kinetic schemes can be to explain these phenomena
and to gain some physical insight, as stated in [24] and [25]. Indeed, it is clear how important it is to
define the input gas mixture, to understand the influence of that mixture on the discharge characteristics
and to understand the kinetics of the various species in the discharge maintenance, which requires the
consolidation of a reference reaction scheme. The experimental work [23] also points out the importance
of the light emissions from N2 (C → B) and N+
2 (B → X) in the ionization front, to follow the evolution of
N2 excited and ion species inside the tube. This emission changes with the N2 concentration, switching
+
from a N+
2 (B) dominated spectrum to a N2 (C)/N2 (B) balanced spectrum when the N2 concentration is
increased from 0.1% to 0.5%. The inclusion of these species in a kinetic scheme is surely important for
comparisons with experimental results.
In [24–28], a two-dimensional discharge code carries out simulations of the propagation of discharges
in a tube filled with static He, containing an admixture of N2 variable from 10 to 10, 000 parts per
million (ppm). The kinetic scheme used in [26] is derived mostly from [29], where a detailed experimental
study of a Patm discharge in He with 6 to 700 ppm of N2 has been done, and from [30], where a twodimensional numerical model of an atmospheric pressure glow discharge in He with some N2 impurities
was developed and successfully reproduced the discharge evolution during the breakdown process observed
+
in experiments. Three positive ions are considered in this scheme (He+ , He+
2 , N2 ), as well as three excited
species (He(23 S)+He(21 S), N2 (C), N+
2 (B)), the latter two for light emission comparisons [26].
Penning reactions and charge transfer reactions, including three-body reactions, are seen as particularly important for discharge ionization and to define the discharge structure, respectively, in Patm
discharges in He with small admixtures of N2 [29]. According to [30], Penning is the dominant ionization
mechanism and the preionization in the case of repeated pulses depends on it, which may define whether
the plasma created is filamentary or a uniform glow. The important role of Penning ionization (He∗
+ N2 → He + N+
2 ) in He-N2 discharges and post-discharges was put forward in many studies, such as
[31–36]. Since it depends on both the N2 and the He∗ densities, it is relevant to know whether and how
the He∗ density changes with the N2 concentration in order to understand the evolution of the Penning
ionization rate with [N2 ]. Unlike [26], bibliographic references [20, 21, 29, 37] state that the He(23 S)
12
excited state can have importance in He-based mixtures as a metastable state by its own, not together
with He(21 S). References [38–40] follow the He(23 S) species evolution, due to its importance for Penning
ionization in several He-based mixtures.
Contrary to the He-N2 mixture, He with a small admixture of O2 , very interesting for medical applications as described earlier, is considered a typical electronegative plasma. According to [31], the modelling
of He-O2 plasma requires many more reactions than that of He-N2 plasmas. Besides electrons and the He
species (metastable atoms, excimer molecules, atomic ions and molecular ions), [31] proposes a reaction
scheme for He with 0.5% O2 at Patm including oxygen atoms O, oxygen molecules O2 , ozone O3 , ions
−
−
−
∗
∗
O+ , O+
2 , O , O2 and O3 and metastable excited states O and O2 . It is claimed that the addition of
various negative ion species not only increases the number of reactions to be considered but also leads to
important structural changes in the plasma. For high electronegativity (n− /ne > 1), there is significant
decoupling between the positive ion and the electron densities. [31] states that the electronegativity decreases and consequently the electron density increases with the power absorbed by the plasma . Figure
I.5 shows the calculated values of the average densities of the most relevant species, as a function of the
power absorbed by the plasma, in a 99.5% He - 0.5% O2 Patm RF micro-discharge [31].
Figure I.5: Density of the most relevant species, versus time-average discharge power, for a 99.5% He 0.5% N2 Patm RF µdischarge [31]. Neutral species on the left and charged species on the right.
Moving forward to He-air mixtures, Naidis (2014) [41] has evaluated the composition of chemically
active species produced in He plasma jets with small admixtures of air (1%-3%) and has identified the
channels for the production of major active species. Humid air was considered, including H2 O, instead
of the more simple dry air model using 80% N2 - 20% O2 . The work adopts the set of reactions for
He-H2 O proposed in [42] (46 species and 577 reactions) and for He-air from [43] (59 species and 1048
reactions). However, among these, particular importance is given to electron-impact-produced primary
+
∗
∗
2
1
1
1
active species He+ , O+
2 , N2 , N, O, He , N2 , N( D), O( D), O2 (a ∆) and O2 (b Σ), created during the
ionization wave propagation. The following are considered as dominating active species: O created by
O2 dissociative collisions with electrons or by N∗2 + O2 →N2 + O + O; O3 created by 3-body association
O + O2 + He → O3 + He; O2 (a1 ∆) created by direct electron excitation; OH created by O(1 D) + H2 O
→ OH + OH; and NO created by N(2 D) + O2 → NO + O. It is particularly noted that the role of
the ions in the plasma jet kinetics is smaller than that of the neutral species N, O and N∗2 . It is also
13
noted that these important active species have density values of about 1013 − 1014 cm−3 (depending on
the operating conditions), but some of them (O and OH) have short life times, quickly decreasing in the
post-discharge relaxation zone, while others (NO and O3 ), with longer lifetimes, accumulate slowly in
the relaxation zone. Even though [41] refers to humid air, the study of He mixtures with dry air can still
benefit from some of its conclusions. A simpler kinetic scheme for mixtures of He with real air (up to
850ppm) is given by [44], resulting in different conclusions, such as the dominance of helium molecular
ions in the positive ion kinetics and the negligible role of negative ions.
The present discussion points out the need to develop and refine kinetic models for He-air jets to
allow further comparisons. In particular, it should be noted that none of the referred models studies the
electron kinetics in detail.
Electron kinetics in Helium with admixture of air components
The majority of the works studying APNP produced in complex gas mixtures do not pay too much attention to the electron kinetics. However, electron-impact collisions are responsible for primarily activating
the discharges and for the creation of the first excited species in the plasma. Knowledge of the EEDF and
of the electron-impact rate coefficients in gas mixtures is useful to the plasma community, as input data
for models, as a validation indicator for comparisons with experimental data (for all types of discharges,
not only transitory jets or Patm plasmas), and as physical quantities of paramount importance for our
understanding of low-temperature plasmas.
Solving the EBE allows us to witness the influence of the gas mixture and the applied field on the
EEDF, considering the particular characteristics of each gas (existence of rotational, vibrational and
electronic excited states; influence of energy thresholds; influence of superelastic collisions; existence of
resonances in the electron cross sections; presence of a Ramsauer minimum in the elastic momentumtransfer cross section). The EEDF provides information necessary to determine the concentrations of the
most important excited species created by electron-impact and to monitor their evolution with changes
in the gas mixture, for instance when varying the concentration of small admixtures of air in the working
gas He. It also allows to promptly study how electrons distribute the power gained from the electric field
by the different collisional processes they are involved in [17]. The effect of small admixtures of molecular
gases is particularly important at low electric fields, in which case the existence of low energy states has
a big influence on the EEDF, even at low concentrations of the admixed gas.
Electron kinetics is well documented for pure gases, as in [12] for He, in [45–51] for N2 or in [49] for O2 ,
and there is easy access to elementary data on electron kinetics for example through the on-line databases
with the platform LXCat [52]. However, for gas mixtures, even though the same electron-neutral cross
sections are used as for pure gases, there is little information on both simulation studies involving the
EBE solution and experimental data for electron-impact rate coefficients. For example, air is the only
gas mixture for which LXCat presents experimental data on the ionization coefficient [53].
Electron rate coefficients calculated from the solution to the EBE for atmospheric-pressure He-N2
plasmas with [N2 ] ≤ 1% are presented in [37]. The following processes were included in the EBE:
excitation of vibrational and electronic states from the ground and the metastable states; ionization by
14
electron impact; superelastic collisions with N and N2 excited states and with the first three He excited
states; and electron-electron collisions. N2 has lower excitation and ionization energies than He, thus the
electron production proceeds mostly through nitrogen in this case. Figure I.6 shows the variation of the
EEDF in a He-N2 mixture with N2 concentrations up to 1%, according to [37]. For N2 concentrations
lower than 1%, the EEDF has the typical features of a pure He EEDF, i.e., quasi-Maxwellian below the
first excitation threshold of He (19.82 eV). At 1% N2 , the EEDF is no longer quasi-Maxwellian, decreasing
for energies above ∼ 2 eV because of the excitation of N2 vibrational modes. This conditions the amount
of electrons available for N2 ionization at 15.5 eV, He(23 S) excitation at 19.82 eV and He ionization at
24.59 eV [12], thus influencing the entire energy workflow ruling the plasma.
Figure I.6: Electron energy distribution function for different percentages of N2 in a He-N2 mixture at
atmospheric-pressure, calculated for a cylindrical container with R = 1 mm and L = 2.3 cm and an
applied power of 600 W (4 < E/N < 8 Td) [37].
Experimentally, the number of references addressing the electron kinetics in He containing mixtures
is also scarce. A Langmuir probe was used in [54] to determine the EEDF in He-N2 mixture plasmas.
In summary, there is a lack of both experimental and modeling detailed information on the electron
kinetics in He-based plasmas, with particular focus on the EEDF main features. This information is essential to provide input data for studying several types of discharges, and gives an important contribution
to understand the overall kinetics of these mixtures.
I.3
Background and starting point
This master thesis is the result of the work developed in both the EM2C (Énergétique Moléculaire et
Macroscopique, Combustion) laboratory at École Centrale Paris and the Gas Discharges and Gaseous
Electronics (GEDG) group of IPFN (Instituto de Plasmas e Fusão Nuclear) at Instituto Superior Técnico
- Universidade de Lisboa. Therefore, it benefits from the competences of both laboratories and from
working with different teams of researchers. This context provides a unique background of research
subjects, expertise and working tools for the development of the thesis.
15
I.3.1
Fluid model for propagation of Helium plasma jets in tubes
At the EM2C laboratory at École Centrale Paris, a two-dimensional discharge code has been developed
for several years to carry out simulations of propagation of discharges in long thin dielectric tubes. This
model can address the study of the influence of N2 on He discharge dynamics in tubes and can provide
qualitative comparison with the experimental results presented in [23].
The set-up consists of a 15 cm long cylindric antisymmetric glass tube (r = 4) of inner radius 2 mm
and outer radius 3 mm. A semi-infinite anode ring is wrapped around the tube in the right-side end.
To ensure that the potential decreases down to zero far from the studied set-up, a grounded cylinder is
placed on the radial boundary at r = 10 cm from the discharge axis and a grounded plane is placed at 100
cm from the set-up left-side end. This guarantees the voltage difference between the anode ring and the
grounded plane, which simulates the potential of air at the exit of the tube. Simulations are carried out
at atmospheric pressure and room temperature, Tg = 300 K and N = 2.45 × 1019 cm−3 , to comply with
experiments. The tube volume where the discharge creates and propagates is filled with static Helium
with an admixture of molecular Nitrogen of 1000 parts per million (ppm) χimp = NN2 /N = 10−3 = 0.1%
[26]. Repetitive voltage pulses are used in plasma jets experiments and the model therefore assumes a
pre-ionization of e− and He+ by previous pulses of ne0 = ni0 = 109 cm−3 and a constant voltage Ua = 6
kV, applied to the anode ring for t ≥ 0. The time-step varies between 10−11 s and 10−12 s.
A classical 2D fluid model is used to simulate the Patm discharge propagation in cylindrical coordinates
(x, r) [26]. Continuity equations (I.1) are solved for the species in the plasma using the drift-diffusion
approximation (I.2) and are coupled with Poisson’s equation (I.3) for the potential V with surface charges
included. The kinetic scheme used is derived mostly from [29] and was mentioned in section I.2.2.
The local field approximation is used, i.e. the transport parameters and electron-impact rate coefficients of the model are functions of the local reduced field E/N . The electron-impact excitation
coefficients and the N2 ionization coefficient are taken from a formula fitted to the tabulated values obtained with the on-line electron Boltzmann equation solver BOLSIG+ [17], using the cross sections from
the MORGAN database of LXCat for He and N2 [52, 55]. The He electron-impact ionization coefficient
is taken from an analytical formula for the reduced Townsend coefficient from [56].
For the drift-diffusion equation, electrons transport coefficients µe and De are also taken from BOLSIG+ [17, 55] with fitted formulas. Positive ion mobilities vary less than 20% with E/N and are taken
as constant average values: µN + = 20, µHe+ = 10.0, µHe+ = 17.5 cm2 V−1 s−1 . Diffusion coefficients for
2
ions are found using the Einstein relation
2
Di
µi
=
kB Tg
qi
with Tg = 300 K.
A general result of this model, for the propagation of a Helium discharge with 1000 ppm of Nitrogen
impurities from the anode towards the cathode inside the tube, can be seen in figure I.7. From this figure,
we see that after 1 µs the discharge was already initiated at the anode (around 100 ns) and propagated
around 5 cm towards the cathode through the tube with very small charge deposition on the dielectric
walls and with average velocity around 5.5 × 104 m/s. On the other hand, a negative polarity discharge
propagates in the other direction much slower and pushed against the dielectric tube. We also notice
that the axial electric field presents a maximum at t =1 µs of 10 kV/cm (at Patm , E/N = 40.8 Td)
in the ionization wave front and a value close to zero in the channel behind the front. However, we see
16
Figure I.7: Simulation of local module electric field, axial electric field, surface charge and electron density
during the propagation of a He discharge with 1000 ppm of N2 at time 1000 ns from the EM2C model.
that the electron density remains high (1012 − 1013 cm−3 ) behind the wave front, showing that there is
a conductive channel between the front and the anode.
In this 2D fluid model, the parameters that directly depend on the N2 concentration are the electron
diffusion coefficient, the electron mobility and the electron-impact reaction rate coefficients. However, at
the starting point of this work, the parameters are fixed for He with 1000 ppm of N2 , which presents a
limitation for studying the effects of the gas mixture on the plasma behavior. In addition, there is no
reference kinetic scheme set for He-N2 atmospheric-pressure plasmas.
I.3.2
Kinetic models for Helium-based discharges
As explained in section I.2.1, the study of the discharge kinetics requires the solution of the rate balance
equations (I.1), coupled with the electron Boltzmann equation (I.7). There is a number of freeware
available codes for solving the EBE, obtaining the EEDF and the corresponding electron transport
parameters and rate coefficients [17, 57, 58], and for calculating the populations of the plasma main
species from the rate balance equations describing their creation and loss [59].
At the Gas Discharges and Gaseous Electronics group of IPFN within Instituto Superior Técnico
(IST), we develop and use an in-house code in MATLAB (R2013a) language called IST-LoKI (LisbOn
KInetics), which keeps the same algorithmic structure and calculation blocks used in [19]. The zerodimensional (0D) CRM couples the particle rate balance equations for the neutral and charged particles
to the EBE (embedding the electron mean-power balance equation), taking into account the elementary
processes considered. In this work, the tool IST-LoKI (LisbOn KInetics) is adapted and used to study
17
the kinetics of atmospheric-pressure He-based plasmas.
As in [60], the input parameters to the model are the electron density, the gas temperature, the
pressure, the oscillation frequency of the electric field and the tube radius for diffusion rate purposes.
The EBE solver module has E/N , the gas mixture and the corresponding electron-neutral collision cross
sections as input data. The homogeneous EBE is solved as described in [17] and in [61]. The EEDF is
expanded in terms of Legendre polynomials of cosθ, θ being the angle between the instantaneous electron
velocity vector v~e and the direction of the electric field [61]. We consider that the electric field and the
collision probabilities have spatial uniformity (local approximation) and we assume small anisotropies,
thus truncating the spherical expansion to first order. The electric field is either stationary or oscillating
but is always considered for a steady-state case, which is why the EEDF can be expanded as a temporal
Fourier series. Electron-electron collisions are not taken into account due to the low ionization degrees
involved (∼ 10−6 ), lower than the onset value of 10−4 , but the possibility for their inclusion exists. The
EBE is written neglecting the production of secondary electrons born in ionization events and the loss of
electrons due to diffusion and recombination, as these mechanisms are expected to have only second-order
effects on the EEDF, at atmospheric pressure. Finally, the EBE is written for the isotropic component
fe0 (ve ) and the anisotropic component fe1 (ve ) of the EEDF as:
1 d 2 0
0
[v (g + ge−0
)] = 0
ve2 dve e E
2 0
1
νc2
eE
dfe
me
kB Tg dfe0
0
0
0
gE
=−
>
0;
g
=
−
v
ν
f
+
<0
e c
e−0
e
3νc νc2 + ω 2 me
dve
M
me ve dve
eE dfe0
1
0
0
~g = (gE
+ ge−0
)e~v ; fe1 = −
νc + jω me νc dve
(I.10)
(I.11)
(I.12)
0
e~v is a radial outward vector in velocity space, since it represents the heating due to the electric
Here, gE
0
field, while ge−0
e~v is directed inward, representing the cooling of electrons due to elastic collisions and
0
0
0
0
0
which can be divided as ge−0
= gel
+ gexc
+ gion
− gsup
, with respect to elastic, excitation, ionization and
superelastic collisions. The product 4πve2 g = G is the flux of the vector ~g through the sphere of radius
ve , called the upflux gain.
The energy space is discretized as a grid, consisting in a series of N subsequent energy intervals
between 0 and umax . The electric field and collisional contributions are set at the N + 1 cell edges, while
the EEDF is defined at the centre of the cells. In this way, the EBE is converted into a set of coupled
algebraic equations, by finite differencing discretization of the original differential equation along the
energy grid, a computational method based on the Rockwood formulation [62]. The electric field and the
electron-neutral collisions contribute as kinetic energy losses or kinetic energy gains for the electrons of
different energies as terms of a matrix. The resulting set of equations is solved by matrix inversion, giving
the EEDF. The transport parameters and the rate coefficients for the particle and energy equations are
then calculated from the EEDF, as in [17].
The solution to the CRM proceeds as follows. First, the EBE is solved for initial-guess values of
E/N and mixture. Then, the set of rate balance equations, for the different neutral and ion species, is
solved using a Runge-Kutta method for ordinary differential equations. The electric field necessary to
18
maintain the discharge is self-consistently determined as an eigenvalue solution to the problem. Since
in steady-state condition the charged-particle net production rate must exactly compensate for the net
loss rate (due to diffusion to the walls and recombination), there is a unique relationship between the
reduced electric field E/N and the product of the gas density and the discharge radius N R. Here, N R
is fixed given R, the atmospheric-pressure and Tg , and hence E/N is the quantity to determine, being
iteratively calculated so as to satisfy the electron ionization-loss balance equation. Since the discharge
provokes changes in the gas mixture composition (e.g. O2 and N2 dissociate into O and N), the mixture
for which the EEDF is calculated also requires self-consistency, under the demand that the EBE is solved
for the final E/N and gas mixture calculated by the chemistry solver. The rate balance equations are
solved for the same E/N and EEDF for the whole temporal evolution until steady-state.
The tool IST-LoKI allows to easily change the gas mixture and to obtain the corresponding transport
parameters and rate coefficients, steady-state densities and E/N . However, at the starting point of this
work, the tool was prepared to solve Argon, Nitrogen and Oxygen mixture systems. In order to use
IST-LoKI for different gas mixtures, the developer must first prepare the code for new electron-neutral
cross section data, new species and new reactions and test the coherence of the solution, by comparing
the results with those from other numerical tools and from experiments.
I.4
Plan of the thesis and original contributions
The plasma community has recently shown interest for Helium-based plasma jets in tubes, due to their
importance in medical applications. The experimental results available on this subject do not allow a deep
understanding of the phenomena involved. Modeling of plasma propagation can be used for comparison
with experimental results and for gaining physical insight into various phenomena, such as the energy
transfer pathways and the species reaction kinetics in the plasma. Given the interest in modeling of
plasma kinetics in Helium-based plasma jets in tubes and the available tools, the priority of my thesis
is to use IST-LoKI to develop collisional-radiative models for mixtures of He with air gases N2 and O2 ,
that can subsequently be implemented in fluid models, such as the one presented in section I.3.1.
The work developed in this master thesis can be summarized as follows:
• Introduction of the electron collisional data for Helium in the EBE solver of IST-LoKI (sections
II.1 and II.2).
This first task requires learning how to work with the code, adapting the data format, developing
the code to meet the requirements of the new species, and testing the results obtained for Helium
by checking the power balance and by comparing simulations with experimental swarm data. The
introduction of the new species allows the study of electron collisions with He ground-state, as well
as with He excited states. It also opens the opportunity for the mixing of He states with N2 and
O2 already present in the LoKI database.
• Study of electron kinetics in Helium-Nitrogen-Oxygen mixtures (section II.3).
Here, the EBE solver of IST-LoKI is used to obtain the EEDF, the swarm parameters and the power
19
terms for each mixture of gases and for each input reduced electric field. The mixture can include
different relative densities of He ground-states, He excited-states, N2 and O2 , but it is focused on
very small He excited-states concentrations (≤ 10−4 ) and on N2 and O2 concentrations between 10
parts per million and 10 %. The effects of mixture variations and E/N variations on the results
are analyzed and conclusions are taken for the study of discharge dynamics.
• Introduction of Helium species and reaction scheme in the chemistry solver of IST-LoKI (section
III.1).
Again, this task requires learning how to work with a new tool and developing the code for a new
purpose. The IST-LoKI is used and developed by several different researchers and therefore the task
has required a coordinated teamwork to be well accomplished. The implementation of the CRM
requires previous bibliographic research and posterior validation with results from the literature.
• Kinetic study of a pure Helium atmospheric-pressure plasma (section III.2).
The first CRM to implement is for pure He. Results are obtained for a steady-state discharge at
Patm and room-temperature with variation of the imposed electron density. Species densities and
reaction rates are analyzed, as well as the calculated E/N . Conclusions of this study can be used
for a scheme reduction for He CRMs. The study also serves as a first step for further study of
He atmospheric-pressure post-discharges, pulsed-field systems and mixtures of He with N2 and O2 ,
which require the introduction of other reactions in the chemistry solver.
• Understanding the influence of N2 concentration on the characteristics of a He discharge with N2
impurities propagating in a long tube at atmospheric-pressure (chapter IV).
The zero-dimensional kinetics solver ZDPlasKin [59] is used with an imposed pulsed field coupled
to the EBE solver Bolsig+ [17], to simulate the local chemistry phenomena in the discharge in the
conditions presented in section I.3.1. By introducing the variation of N2 concentration to the 2D
discharge code, we can obtain all the modeling results for different He/N2 mixtures. The zeroD results for the kinetics are coupled with the 2D results of the discharge dynamics in the tube.
Consolidation of the reaction scheme from references is necessary to better understand the influence
of N2 concentration in the dynamics of the discharge.
The work developed in this master thesis has contributed both to the advancement of science and
to the development of the IST-LoKI tool, resulting in contributions published in several proceedings of
international conferences [24, 27, 28, 60, 63, 64].
20
Chapter
II
Electron kinetics in Helium-based plasmas
II.1
General formulation for the numerical solution of the electron Boltzmann equation
Following section I.3.2, the tool IST-LoKI (LisbOn KInetics) is adapted and used to study electron-impact
reactions in the aforementioned conditions with He. IST-LoKI solves the Lorentz two-term expansion of
the homogeneous steady-state electron Boltzmann equation (I.10). The study focuses on the EEDF of
helium-containing plasmas, calculated for several values of the reduced electric field (E/N ), analyzing the
effects of small admixtures of N2 , O2 and synthesized air, and the influence of He metastables involved in
stepwise ionization and superelastic collisions. With that intent, three cross section files were created to
serve as input to the numerical solver, using the cross section data from Santos et al. (2014) [12]. One
of these files contains cross sections for the elastic, electronic excitation and ionization collisions between
electrons and ground-state He (He(11 S), statistical weight g = 1). The other two files contain cross
sections for the elastic, electronic excitation and ionization collisions between electrons and metastable
excited states He(23 S) (g = 3) and He(21 S) (g = 1).
The EEDF calculation allows to obtain the transport parameters , the ionization coefficient obtained
from eq. (I.8), the reduced Townsend ionization coefficient (α/N ) and the power balance components,
for several values of the reduced electric field and the relative density of species. The results are validated
by comparison of swarm parameters with other numerical results and with some experimental results.
The consistency of the EBE resolution is verified by the electron energy conservation. In fact, the EBE
(I.10) embeds the electron mean-power balance given by [12]:
PE + Psup = Pel + Pexc + Pion
(II.1)
The power-balance eq. (II.1) is obtained by multiplying eq. (I.10) by 4πve2 to obtain the flux, then
by the electron energy
ve2 me
2 ,
and integrating over all energies/velocities. The terms on the left-hand side
of eq. (II.1) represent, in order, the mean power absorbed from the field per electron PE and the mean
power gained in superelastic collisions Psup , whereas the terms on the right-hand side represent the mean
21
power lost in elastic collisions with ground-state and excited-state atoms Pel , in excitations Pexc and in
ionizations Pion , respectively. Energy conservation requires the gain and loss terms to compensate each
other. Each power term is the integration of the corresponding term in eq. (I.10). Detailed expressions
for the different power terms can be found in [20]. In particular, the power transfer in inelastic and
superelastic collisions from state i towards state j, Pij is calculated by multiplying the energy threshold
of the process, uij , by the rate coefficient of that reaction, Kij , and the relative density of state i:
Pij = uij × Kij × ni /N .
II.2
II.2.1
Validation of collisional data
Solution for Helium plasmas
The first step required in the task of studying Helium is the inclusion of its collisional data in the program
database and the validation of the corresponding results. When we consider pure He, the Boltzmann
equation is solved only for the e− + He(11 S) collision cross sections from [12] and there is no effect of superelastic collisions. These cross sections are for elastic collisions, inelastic collisions towards all the states
He(n2s+1 l) with n < 5 and the ionization He → He+ collision. They were obtained from the set proposed
by [65] and adjusted to ensure good agreement between calculated and measured swarm parameters [12]
and can be read in an online supplementary data file (stacks.iop.org/JPhysD/47/265201/mmedia). We
can see in figure A.1 of appendix A the cross sections of elastic collisions between electrons and groundstate He, of excitation collisions e− + He (11 S) → e− + He (23 S) with energy threshold 19.82 eV and of
ionization collisions e− + He (11 S) → He+ + 2 e− with energy threshold 24.59 eV. Linear interpolation
was used always. We notice in figure A.1 that there are input σ values between 0.2 eV and 1000 eV. The
interpolated cross sections used in the calculations need to be correctly discretized in order to exclude
unreliable values and ensure a correct power balance. As expected, elastic collision cross sections have no
energy threshold and no change in order of magnitude until approximately 10 eV. Then, they decrease
three orders of magnitude until u = 1000 eV. The excitation and ionization cross sections have the same
shape, although with different energy thresholds and energy maxima. The cross section values for the
ionization collision are higher than the ones for excitation at high energies, up to 5 orders of magnitude.
Figure II.1 shows the results of the EEDFs calculated by the Boltzmann equation solver and the
u(eV )
results of a Maxwellian distribution in thermal equilibrium (fM B = C × e− Te (eV ) , Te = 23 ū), that uses the
average energy ū calculated by the Boltzmann solver, for an energy range until 200 eV and for several
values of E/N from 1 Td until 1000 Td. In figures A.2 and A.3 in appendix A, the same plots are
presented with more detail, comparing the calculated and the Maxwellian distributions for each E/N .
As mentioned before, excitation of ground-state Helium by electron collision has a first energy threshold of 19.82 eV and the ionization of this state has a threshold of 24.59 eV. From figures II.1,A.2,A.3,
we can notice that for E/N = 1 Td, the amount of electrons that reach the threshold energies will be
infinitesimal. However, for E/N = 10 Td or 50 Td, we can see that electrons are restrained by these
inelastic collisions as the EEDFs are quasi-Maxwellian until energies near the thresholds, being depleted
for higher energies. For 100 Td, the electrons will not only be able to provoke excitations and ionizations
22
Figure II.1: Results for the EBE-calculated (left) and Maxwellian (right) EEDFs for pure ground-state
Helium in logarithmic scale at different reduced electric fields.
but a reasonable amount can be accelerated until hundreds of eV. At E/N higher than 250 Td, we notice
a plateau at high energies, which means electrons are not losing enough energy by collisions to lower the
EEDFs. At 1000 Td, for instance, we notice a plateau near 5 × 10−5 eV−3/2 .
The swarm parameters are calculated by LoKI from the EEDFs, as function of E/N . The values
of the reduced electron mobility µe × N , the electron drift velocity vde , the reduced electron diffusion
coefficient De × N , the electron characteristic energy uK and the reduced Townsend ionization coefficient
α/N are calculated as in [17] with the possibility of including High-Frequency fields. Using f = 0, these
calculations show zero imaginary part and a real part equal to the DC calculated values, which shows
coherence. The calculations follow the expressions:
p
Z
2e/me +∞ u dfe
du
3
σef f du
0
p
Z
2e/me +∞ u
De × N =
fe du
3
σef f
0
µe × N = −
vde = µe × E; uK = De /µe ; α/N = Kion /vde
(II.2)
(II.3)
(II.4)
The same parameters can be calculated from the power gained by the electrons from the electric field
and we notice that they increasingly diverge from the precedent calculations for E/N > 200 Td, up to
several orders of magnitude.
vde =
vde
PJ
; µe =
E/N
E
uK = De /µe ; α/N = Kion /vde
(II.5)
(II.6)
As expected, the power gained by electrons from the electric field increases with E/N . The power
lost by electrons in elastic collisions increases also with E/N , since a higher amount of electrons have
kinetic energies for higher cross sections. As expected, the power lost by electrons in inelastic excitation
and ionization collisions increases with E/N due to the shape and energy thresholds of the inelastic cross
sections. Finally, the differences between gained and lost power are well balanced, except for E/N > 200
23
Td. This result is coherent with the results in figure II.1 for high E/N , where the EEDF never drops
from a certain level for high u, which suggests there may not be enough losses through collisions at high
u to compensate the acceleration by the electric field. Therefore, the differences between the two ways of
calculating swarm parameters are due to the calculation of the power terms. The Boltzmann solver does
not include a term of creation of secondary electrons by electron impact ionization, with the consequent
electron energy sharing in the EEDF calculation, which would be more important for high E/N fields
and would affect the shape of the distribution function and the calculation of the swarm parameters and
power transfer terms. That is the explanation for the inconsistency of the EEDFs and swarm parameters
at E/N > 200 Td.
The swarm parameters De ×N , µe ×N , uK and α/N , calculated for E/N between 0.01 Td and 300 Td
and for the case of pure ground-state He, were compared to those obtained from the on-line Boltzmann
solver Bolsig+ [17], using the cross section databases of IST-LISBON [66] and 3 different models (no
energy sharing, pulsed Townsend (PT) model and steady-state Townsend (SST) model). They were also
compared to the results published in Santos et al. (2014) [12], that used a different code and additional
cross sections, with excitations towards levels n2S+1 l, 4 ≤ n ≤ 7, and to experimental values available
on the LXCat on-line platform [52, 53, 66, 67]. These comparisons allow us to evaluate the validity
of the EEDF calculation in LoKI, as well as the calculation of transport parameters and reaction rate
coefficients like Kion , since α/N = Kion /vde , and they are presented in figures II.2 and II.3.
Figure II.2: Swarm parameters for pure He(11 S) as function of E/N , obtained from several models and
experiments: reduced electronic diffusion coefficient on the left and reduced electronic mobility on the
right.
As stated in [12], there is good agreement of the calculated swarm parameters with experiments for
E/N ≤ 100 Td. However, the problem witnessed for the power balance at E/N ≥ 200 Td is also revealed
here as an overestimation of the swarm parameters for 100 Td < E/N ≤ 300 Td. The Bolsig+ SST spatial
growth model with equal sharing of energy shows convergence only until 344 Td. Up to that value, the
results seem slightly better fit to the experimental values than those of the PT temporal growth model
with equal sharing of energy. The Bolsig+ no energy sharing model shows convergence only for E/N ≤
236 Td. We can notice that the LoKI results agree with those of [12], which means that the inclusion of
further excitation cross sections towards levels n2S+1 l, 4 ≤ n ≤ 7, present in [12], would not provoke a
24
Figure II.3: Swarm parameters for pure He(11 S) as function of E/N from several models and experiments:
reduced ionization Townsend coefficient on the left and electron characteristic energy on the right.
significant change in the calculated EEDF.
Comparing the LoKI results with the entire set of experimental results, it has been verified that the
difference between the LoKI result for De × N and the closest experimental De × N at E/N = 100 Td is
8.75 %. For µe × N , the difference between the LoKI results and the closest experimental result is 1.36
% and 11.55 % at E/N = 100 Td and E/N = 300 Td, respectively. At the same fields, the difference
between results for the characteristic energy is 8.37 % and 39.37 %, respectively. The results of α/N
using Bolsig+ without energy sharing have perfect agreement with those using LoKI and those in [12].
However, the Bolsig+ energy sharing SST model is the one that agrees best with experimental results of
α/N for E/N > 100 Td. At E/N = 100 Td, the difference of α/N calculated by LoKI and the closest
experimental result is 27.16%, while at E/N = 300 Td it is 45.89 %. The reduced electric fields taken as
reference are ∼ 5 Td from [12] and ∼ 40 Td from section I.3.1, always between 1 Td and 100 Td, and
therefore the results presented in this section validate the calculations for Helium presented hereafter.
II.2.2
Solution for Nitrogen, Oxygen and dry air plasmas
To study mixtures of He with N2 and O2 , data from N2 and O2 needs also to be validated. The input
electron-neutral collision cross sections for N2 and O2 used in LoKI are the same as in [66]. These
include data for elastic, rotational excitation, vibrational excitation (first energy threshold 0.29 eV),
electronic excitation (6.17 eV) and ionization (15.5 eV) collisions with N2 and elastic, rotational excitation,
vibrational excitation (0.19 eV), electronic excitation (0.98 eV), ionization (12.1 eV) and attachment
collisions with O2 . In this context, figure II.4 represents the electron kinetics parameter that matters
the most in the study of ionization mechanisms, and that can be compared with experimental data, the
reduced Townsend ionization coefficient α/N = Kion /vde . Swarm parameters De × N , µe × N and uK
are represented for the same conditions in figure A.4 in appendix A. The results obtained from LoKI are
compared to the results obtained from the several Bolsig+ models, using the IST-Lisbon cross section
data [66], and from experimental data retrieved in [52], for an E/N range until 1000 Td.
For N2 , figure II.4 reveals good agreement between the models and the experimental values, except for
25
Figure II.4: Reduced Townsend ionization coefficient for N2 (left) and O2 (right) as function of E/N
from several models and experiments.
E/N ≥ 300 Td, in which case the energy sharing models present better agreement. For O2 , without access
to experimental values for α/N at E/N ≥ 300 Td, all the models present an equivalent agreement with
experiments. The calculations obtained for theses gases are therefore validated for ulterior utilization.
The gases N2 and O2 have particular interest as part of atmospheric air. We can check the validity
of considering the approximation of dry air, composed of 80% N2 and 20% O2 , by comparing results
from LoKI using this mixture with experimental data obtained from pure air. As mentioned in section
I.2.2, the only experimental data concerning ionization in the LXCat platform is the one from the Dutton
database [53]. Unlike the experimental data obtained for O2 , the data presented for air concerns an
effective ionization Townsend coefficient, calculated by subtracting the electron-impact attachment rate
of Oxygen (e− + O2 → O− + O) from the electron-impact ionization rate of both Nitrogen and Oxygen
α
−
(N
η
N
=
KN 2+ ×0.8+KO2+ ×0.2−Katt ×0.2
).
vde
The solver LoKI allows us to obtain the same data, since the
input collisional data includes both the ionization and the attachment reactions. Figure II.5 allows to
α
compare the results from LoKI, for both the ionization coefficient alone ( N
=
KN 2+ ×0.8+KO2+ ×0.2
)
vde
and
the effective coefficient, with the experimental data from [53].
Figure II.5: Reduced Townsend effective ionization coefficient, for air from experiments and for dry air
from LoKI (LoKI effective), and reduced Townsend ionization coefficient (not effective) for dry air from
LoKI (LoKI ionization), as function of E/N .
26
From figure II.5, we realize that there is good agreement between the numerical and the experimental
values between 200 Td and ∼ 500 Td. The numerical result is higher than the experimental result at
higher E/N , in coherence with the results from figure II.4. However, the subtraction of the calculated
attachment coefficient from the ionization coefficient leads to a negative effective ionization coefficient
below 100 Td, while the experimental values are already positive at 100 Td and present a better agreement
with the calculated ionization coefficient alone. This probably means that it is experimentally difficult
to measure ionization or attachment coefficients alone, since the phenomena affecting the number of
electrons - ionization, attachment and detachment - are all happening simultaneously. Overall, the dry
air approximation seems to correctly represent air as far as the electron-impact ionization coefficient is
concerned. Comparisons for swarm parameters µe × N and uK are presented in figure A.5.
II.3
Mixtures of ground-state Helium, Helium excited states,
Nitrogen and Oxygen
II.3.1
Influence of Helium excited states on the electron kinetics
It was mentioned in section I.2.2 that some He metastable states have long lifetimes. This means that
they will not disappear quickly by radiative decay and, conversely, will continuously collide with electrons
and influence the EEDF. Among the metastable states, the ones with principal quantum number n=2,
He(23 S) and He(21 S), are the ones with most relevant densities. According to [12], their relative densities
with respect to the ground-state densities are expected to be between 10−9 and 10−6 . Therefore, they
should be considered when sloving the EBE. When a relative density is set for the excited states He(23 S)
(statistical weight g = 3) and He(21 S) (g = 1), electrons also collide with these states and the collision
cross sections take part in the calculation of the EEDF. In practical terms, these metastable states are
introduced as different species, meaning that the gas is now a mixture of ground-state He(11 S) and
excited states He(23 S) and He(21 S). Elastic collisions are considered between electrons and all the
three states, using the same cross sections. In addition, the new collision cross sections include inelastic
excitations and ionization from these excited states to upper levels and superelastic collisions, bringing
the metastable states back to lower levels. The superelastic electron cross sections are calculated using
the Klein-Rosseland relation:
σji (u) =
σij (u + ∆uji )(u + ∆uji ) gi
u
gj
(II.7)
where u is the electron energy, ∆uji ≡ uj − ui > 0 is the energy threshold for the transition from level i
to level j and gi and gj are the statistical weights of the lower and upper levels, respectively.
The most relevant electron cross sections for the ensemble of all three He species are represented
in figure A.6 in appendix A. We can notice in that figure the introduction of stepwise inelastic cross
sections, of which the most important are the He(21 S) → He(21 P ) and the He(23 S) → He(23 P ), with
energy thresholds of 0.602 eV and 1.144 eV, respecticely, and values higher than 10−18 m2 between 2 and
7 eV and between 1 and 17 eV, respectively. Besides, all the energy thresholds of the stepwise inelastic
27
collisions have values lower than 5 eV. Thus, the introduction of these collisions will have a bigger effect
on the EEDFs at energies below 5 eV. Looking at the EEDFs in figure II.1, we see that the effect at low
energies will probably have a higher influence at low E/N = 1 Td, the first case to study.
Before studying the effect of the introduction of all the inelastic and superelastic processes from the
excited states, we first try to understand what is the effect of each process to include. On the left-hand
side of figure II.6, we study the case of ground-state He with all the e− + He(11 S) cross sections and
the addition of only one superelastic process from an excited state towards the ground-state (He(23 S) →
He(11 S)), for different relative densities of He(11 S) and He(23 S). On the right-hand side of figure II.6,
all the inelastic processes between electrons and He(23 S) are included, in addition to those previously
considered. The values of He(23 S) relative densities for which the EEDFs are being studied in figure II.6
are not necessarily physical, since there is no expectation of values higher than 10−6 . Although, they are
helpful to gain physical insight into the new phenomena.
Figure II.6: Results for the calculated EEDFs for E/N = 1 Td and several mixtures of He(11 S) and
He(23 S). On the left, only e− + He(11 S) processes and superelastic He(23 S) → He(11 S) process are
included. In the legends He(x, y, z), x, y and z are the relative densities of He(11 S), He(23 S) and
He(21 S). On the right, all electron-impact processes from He(11 S) and He(23 S) are included.
Figure II.6 shows that the inclusion of the superelastic process alone can have a huge effect in the shape
of the EEDF. In fact, by increasing the density [He(23 S)], we are increasing the proportion of electrons
performing superelastic collisions that do not require energy threshold and that provide an energy gain
of 19.82 eV to the electron. This energy gain explains the plateau observed between approximately 0
and 20 eV. This plateau effect had already been reported in [17, 46, 48]. Consequently, fe is lower for
very low energies 0-1 eV. Coherently, the increase of 1 order of magnitude in the relative density of
the metastable state results in approximately 1 order of magnitude of difference in the value fe of the
plateau. The addition of the inelastic processes also plays a role on the EEDF shape. We notice on the
right-hand figure II.6 that fe now is higher at very low energies (0-1 eV) and that there is a deep fall
until ∼ 3 eV, in the region where the new inelastic cross sections matter the most. After the plateau
region, there is once again a drop of fe due to inelastic collisions with both He(11 S) and He(23 S). The
higher the metastable densities, the deeper the falls of fe . If we expand the scale of energies, we see that
there is a continuous cycle of drops and plateaux of ∼ 20 eV, due to inelastic and superelastic processes,
28
respectively. Figure A.7 in appendix A presents the same physics as the right-hand side of figure II.6 but
for the cases with He(21 S) densities instead of He(23 S) and the case with both metastable states with
equal relative densities. The effect of long plateaux due to superelastic collisions and deep falls due to
inelastic collisions is the same. However, in this case we consider three superelastic collisions He(21 S)
→ He(23 S), He(23 S) → He(11 S) and He(21 S) → He(11 S) with threshold values of 0.80 eV, 19.82 eV
and 20.62 eV, respectively. In summary, the addition of these relative densities provides more energetic
EEDFs, due to the effect of superelastic collisions. We must take into consideration that ionization is
obtained for electron energies higher than 3.97 eV for collisions with He(21 S), 4.77 eV for collisions with
He(23 S) and 24.59 eV for collisions with ground-state He. Therefore, for metastable relative densities of
10−7 , as the ones in figure II.6, ionization may already be important, even for such low E/N .
It is important to study the effect of the introduction of metastable state densities also for higher E/N
fields, e. g. in the range 10-100 Td, as reported in section I.3.1. In figure II.7, it can be observed how the
EEDFs change with the relative densities of the ground state and the metastable states for E/N = 10 Td
and E/N = 50 Td, under the condition [He(23 S)]=[He(21 S)]. For these E/N s, the EEDF seems to have
a less energetic body but a more energetic tail when the metastable states are added. Comparatively to
the case of 1 Td, inelastic collisions that require an energy threshold happen more often and superelastic
collisions that have no such requirement lose importance.
Figure II.7: Results for the calculated EEDFs for E/N = 10 Td (left) and E/N = 50 Td (right) and
several mixtures of He(11 S), He(23 S) and He(21 S) with [He(23 S)] = [He(21 S)].
Unlike the E/N = 1 Td case, here a relative metastable state density of 10−5 is needed to change
the EEDF shape for the 10 Td case, increasing to 0.01% for the 50 Td case. However, the shape of clear
drops and plateaux is visible only for metastable relative densities higher than 10−4 for the 10 Td case
and 1% for 50 Td. On the other hand, as for E/N = 1 Td, the EEDF accumulation at low energies
for high metastables densities is due to the inelastic collisions with He(23 S) and He(21 S), that happen
at energies starting from 0.348 eV. The other EEDF accumulation near 20 eV happens thanks to the
electrons that perform superelastic collisions with He(23 S) and He(21 S), acquiring 19.82 eV and 20.62
eV, respectively. The case for E/N = 100 Td, not represented here, has shown the same trends as the
one for 50 Td. In conclusion, even though the collisions between electrons and the excited states change
the EEDFs at these values of E/N , the global effect is less visible than at E/N =1 Td, specially since
29
relative densities of metastable states are expected to be lower than 10−6 .
Since the effects of the inclusion of the excited states densities on the EEDF have already been
studied, we now seek to understand how the changes in the EEDF affect the calculation of the swarm
parameters and the power balance. As expected, the inclusion of superelastic collisions leads to the
introduction of a new term of power gain. But the introduction of further excitation collisions and of
lower threshold ionization collisions also leads to higher power losses. There is more power lost by electrons
in inelastic collisions and more power gained by electrons in superelastic collisions. To counterbalance, as
the metastable states relative densities increase, the power lost by electrons in elastic collisions decreases.
All the results presented in this section have shown a relative power balance tending towards zero, which
means that equation (II.1) was verified and that the approach of introducing excited states as separate
species from the ground-state is consistent with eq. (II.1).
Figure II.8 presents the electron-impact ionization coefficient Kion and the reduced Townsend ionization coefficient α/N for varying ground-state He and He(23 S) densities (between 0 and 10−4 ), calculated
for a range of E/N between 0.1 Td and 300 Td. For the same relative densities and reduced fields, figure
A.8 in appendix A presents the swarm parameters De × N , µe × N and uK . In figure A.8 we notice
that only high relative densities of metastable states, like 10−4 , cause significant changes in the swarm
parameters. This means that the changes provoked on the EEDF by the addition of excited states with a
relative density of 10−6 , observed in figures II.6 and II.7, do not have a significant effect on the diffusion
and mobility coefficients.
Figure II.8: Global ionization coefficient (left) and reduced Townsend ionization coefficient (right) as
function of E/N for different mixtures of He(11 S) and He(23 S).
Focusing again on the ionization coefficients, we notice in figure II.8 that even for the lowest fields,
below 1 Td, stepwise ionization always exists, since it requires very low energy thresholds (4.77 eV and
3.97 eV). Therefore, the weighted ionization coefficient increases as we add more excited-state densities.
The effect of the stepwise ionization rates on the global ionization had already been reported in [17, 47].
For high densities of He(23 S) (10−4 ), stepwise ionization is relevant until ∼100 Td, whereas for low values
of He(23 S) concentrations, such as 10−6 , its effect is relevant at low E/N < 2 Td, even with low absolute
value (Kion ∼ 5 × 10−24 m3 /s at 1 Td and ∼ 10−19 m3 /s at 10 Td). However, between 2 Td and 20
Td, stepwise ionization appears to give a significant contribution to the global ionization coefficient, even
30
for low relative densities of metastables, like 10−6 . This is the direct consequence of the plateaux in the
EEDFs shown in figure II.6 that increase fe at the ionization threshold energies. The results for the
Townsend ionization coefficient reflect the same effects. At low E/N , with constant µe × N , the electron
drift velocity vde = µe × E decreases as E/N decreases. Since at low E/N the ionization coefficient
Kion remains constant, the corresponding reduced Townsend coefficient α/N = Kion /vde increases as
E/N decreases. Therefore, the presence of excited states leads to a minimum of α/N when Kion starts
increasing due to ground-state direct ionization, between 2 Td and 5 Td.
Results of swarm parameters De × N , µe × N , uK , α/N and Kion obtained for E/N between 0.01
Td and 300 Td and for the cases of pure ground-state He and He with relative densities of 10−4 for
metastable states He(23 S), He(21 S) and both are presented and analyzed in figure A.9 in appendix A.
It is not usual to calculate or measure swarm parameters for excited-state gases and, therefore, figures
II.8, A.8 and A.9 present no comparison with other results. In summary, the inclusion of He metastable
states and their reactions with electrons for the calculation of the EEDF and the swarm parameters is
very important for reduced electric fields lower than 30 Td.
II.3.2
Influence of N2 , O2 and dry air admixture to Helium on electron kinetics
The solution of the Boltzmann equation for He, N2 , O2 and dry air, as well as for He with various
densities of metastable states, has been presented and in some cases validated in the previous sections.
Electron kinetics for mixtures of these gases can now be studied, analyzing the influence of the mixture
composition on the electron kinetics. Once again, we start by studying the EEDFs for different reduced
electric fields, this time varying the density of each gas. All mixtures can be studied, but given the
conditions presented in chapter I, focus is given to the cases of He-based plasmas with small densities of
N2 , O2 and dry air (80% N2 - 20% O2 ). Figure II.9 presents the EEDFs calculated for E/N = 10 Td
(between the values ∼ 5 Td of [12] and ∼ 40 Td of section I.3.1) for different mixtures He-N2 on the
left and He-O2 on the right. Helium is used here in pure ground-state and the distribution of vibrational
states in N2 is obtained through the Treanor distribution at 300 K.
We can observe in figure II.9 that He presents a very different and more energetic EEDF than N2
and O2 . The excited and ionized states of He have very high energy thresholds, in contrast with the
low-energy electronic and vibrational excited states of N2 and O2 , leading to a depletion of the EEDF
tail when these gases are admixed. In addition, we observe that even small admixtures of the molecular
gases, like 1%, produce a significant change in the EEDF. The result is similar to the one reported in
[37] and in section I.2.2. By increasing the amount of the molecular gas, we see that the EEDF of the
mixture approaches that of the molecular gas at low energies, which means that the low-energy threshold
inelastic collisions with N2 /O2 are dominating in this region. Although the EEDFs of N2 and O2 have
different shapes, due to the different excited states and energy thresholds of these gases, their effect on
the EEDF when admixed with He at relative densities up to 1% is somewhat similar.
The left-hand figure II.10 presents EEDFs calculated for He-air mixtures, for the same conditions as
in figure II.9. As expected, the result is a combination of the results of figure II.9, presenting the same
31
Figure II.9: Results for the calculated EEDFs for E/N = 10 Td and several mixtures of He(11 S) and N2
(left) and of He(11 S) and O2 (right).
phenomena. Figures A.10, A.11 and A.12 in appendix A compare EEDFs for different He-N2 , He-O2
and He-air mixtures, respectively, for E/N = 1 Td (extreme case below the low reference field ∼ 5 Td
[12]) and E/N = 50 Td (same order of magnitude as the maximum field ∼ 40 Td of section I.3.1). The
results in these figures show the same trends as the results at 10 Td. However, we notice that at higher
fields (50 Td) the effect of small admixtures is less noticeable. For example, unlike the case at 10 Td, it
is not observed the slight depletion of the EEDF for an admixture of 0.1% of the molecular gas. On the
contrary, the case at 1 Td presents clear differences in the shape of the EEDFs at low energies, where the
depletion is more noticeable for 99.9% He - 0.1% N2 than for 99.9% He - 0.1% O2 . Moreover, while the
EEDF for the case of 50% He - 50% O2 seems to approach the EEDF for pure O2 , the EEDF for 50%
He - 50% N2 looks divided between He and N2 .
On the right of figure II.10, we observe the results of EEDFs when He with relative densities 10−4
of excited states He(23 S) and He(21 S) is mixed with O2 . We should remind that 10−4 is a too large
value for the relative densities, but it exemplifies well the effect of the metastable states. The results
Figure II.10: Results for the calculated EEDFs for E/N = 10 Td and several mixtures of He(11 S) and
dry air (80% N2 - 20% O2 ) (left) and for E/N = 1 Td and several mixtures of He(11 S), He(23 S), He(21 S)
and O2 , with He(23 S) and He(21 S) relative densities of 10−4 (right).
clearly reveal different phenomena than those studied earlier. We can see that the increase in the O2
32
concentration leads to the emergence of ‘peaks’ in the region of the plateau created by He superelastic
collisions. In fact, the peaks in figure II.10 are evidence of the inelastic electron-O2 collisions of energy
thresholds 0.19, 0.38, 0.57, 0.75, 0.98, 1.63, 4.5, 6.0, 8.4, 9.97 and 14.7 eV. They are not noted when there
is only ground-state He and O2 at E/N = 1 Td because the EEDFs quickly decrease, which means that it
is possible to notice these ‘peaks’ only in the presence of the plateau effect. The persistence of the peaks
in the energy region 2-22 eV is due to the ineffectiveness of elastic electron-neutral collisions in smoothing
the EEDF, in the case where the plateau exists. Figure A.13 in appendix A shows the results of calculated
EEDFs at 10 Td and 50 Td for He-O2 mixtures, considering excited He with relative densities 10−4 of
He(23 S) and He(21 S). At these values of E/N , the EEDFs have values higher than those of the plateau
and the ‘peaks’. That is why they are not visible at 50 Td and they are less distinguishable at 10 Td
than at 1 Td. In fact, at 50 Td the results are similar to the ones obtained without the presence of the
metastable states.
Once the EEDFs have been calculated, several parameters can be obtained for the various mixtures,
for a wide range of reduced electric fields. The changes observed for the EEDF induce differences also
in the calculated rate coefficients, particularly in the global electron-impact ionization coefficient Kion .
Figure II.11 shows this coefficient for He-N2 mixtures
Kion =
KHe+ [He] + KN 2+ [N2 ]
[He] + [N2 ]
Here, the evolution with E/N of the coefficients is compared for several He-N2 mixtures, in the range
of interest (1-100 Td). Moreover, the contribution of N2 ionization in the global ionization coefficient
(KN 2+ × [N2 ]/N ), obtained for chosen values of E/N of 5 Td (from [12]), 20 Td and 60 Td (intermediate
values) is observed as function of the relative density of the N2 species in He-N2 mixtures.
Figure II.11: Global electron-impact ionization coefficients, as function of E/N (left), and contribution
of N2 electron-impact ionization as function of the relative density of N2 , for three values of E/N : 5 Td,
20 Td and 60 Td (right).
As predicted, the depleted tail of the N2 EEDF with respect to the pure-He EEDF leads to a much
lower ionization coefficient in N2 than He, in the range of E/N considered. Results show that the
admixture of small percentages of N2 has little effect on the ionization coefficient, in coherence with the
minor changes caused also in the EEDF (see figure II.9). But although the distribution functions are
33
always more energetic for pure He than for mixtures, as seen in figure II.9, we observe on the left figure
II.11 that a maximum of the ionization coefficient is obtained for relative densities of N2 close to 0.1%,
for fields below approximately 20 Td, while a large decrease is obtained already at 10% N2 for the whole
range of E/N . In fact, the lower ionization threshold of N2 explains how can EEDFs similar to those
of pure He yield higher ionization coefficients if only a small quantity of N2 is added. This means that
the coefficients for electron-He reactions (excitations and ionization) always decrease with the addition of
N2 (although very slowly until ∼ 10% N2 ) whereas, as shown on the right figure II.11, the contribution
of N2 for direct ionization shows a maximum with the increase in the N2 concentration (near 0.1% for
5 Td, near 1% for 20 Td and near 10% for 60 Td) and then decreases thanks to the depletion of the
EEDF, which explains the result obtained for the global coefficient. The overall results obtained for
He-N2 mixtures suggest that a small admixture of the molecular gas to He may be a helpful feature in
order to create and maintain the discharge.
The other important binary mixture to consider is He-O2 . The effect in the EEDF of an admixture of
O2 is similar to the one observed for He-N2 mixtures. However, the electronegativity of O2 is shown to
have an important effect on the plasma behavior. The inclusion of oxygen brings about the introduction
of an electronic attachment process, as mentioned in sections I.2.2 and II.2.2, that allows calculating
an effective ionization coefficient, subtracting the attachment rate coefficient from the direct electronimpact ionization rate coefficient. On the left of figure II.12, we observe the effect of that subtraction by
comparing the effective coefficient in the range of E/N 1-100 Td for several He-O2 mixtures, considering
He ionization, O2 ionization and O2 attachment, with focus in small concentrations of O2 . On the right
side of the same figure, the evolution of the global ionization coefficient with O2 relative density, without
considering the attachment process, is compared for the same three values of E/N used in figure II.11.
Figure II.12: Electron-impact effective ionization coefficients, as function of E/N (left), and electronimpact ionization coefficient, as function of the relative density of O2 , for three values of E/N : 5 Td, 20
Td and 60 Td (right).
Firstly, we verify in the left figure that the bigger the quantity of O2 in the mixture, the higher is the
electric field for which the ionization rate coefficient exceeds the attachment rate coefficient and a positive
effective coefficient is obtained, revealing a possible difficulty for discharge breakdown. In fact, the curve
for pure O2 can not be seen in the figure, since the effective coefficient is only positive for electric fields
34
higher than 100 Td. This is the major difference for the electron kinetics, between admixing oxygen
instead of nitrogen. The remaining phenomena are similar to the ones observed for He-N2 mixtures. In
fact, the maxima of the global ionization coefficient are obtained for molecular gas quantities between
0.1% and 1% in both cases. On the right figure II.12, where the attachment process is neglected, we
notice that for low field E/N = 5 Td the global ionization is maximum near relative density of 0.1% O2 ,
while for an intermediate field E/N = 20 Td the same happens near 1% O2 and for a higher field E/N
= 60 Td, a maximum is not even noted. Once again, similarly to He-N2 , the excitation and ionization
coefficients for electron-Helium reactions decrease with increasing oxygen relative density. However, the
contribution of O2 to the global ionization KO2+ × [O2 ]/N first increases with [O2 ] until it reaches a
maximum, which explains the results obtained for the global coefficient. Similarly, the O2 excitation
coefficient KO2exc × [O2 ]/N also changes with the oxygen relative density. For instance, the excitation
coefficient towards dissociative states of energy thresholds 6 eV and 8.4 eV presents maxima for values
near 2% O2 for 5 Td, 5% O2 for 20 Td and 100% O2 for 60 Td. All things considered, although oxygen
carries a difficulty for discharge creation, a very small admixture seems to have the same effect of adding
N2 , thus favoring ionization in the plasma.
Having studied the binary mixtures He-N2 and He-O2 , it is possible to consider the study of the
ternary mixture of Helium with dry air (80% N2 - 20% O2 ). We have concluded from figure II.10 that
the He-air EEDFs are very similar to the He-N2 EEDFs. However, the electronegativity of O2 is shown
to have an important effect on the plasma behavior, as it adds an attachment reaction to the electron
kinetics. Figure II.13 depicts the effective ionization coefficient, defined as the difference between the
electron-impact ionization coefficient and the attachment coefficient, calculated for several mixtures of
He with synthesized air (80% N2 - 20% O2 ) and for the E/N range 1-1000 Td. For the electron kinetics,
the main difference between admixing nitrogen and dry air to Helium is, in fact, the presence of the
electronic attachment reactions of oxygen, responsible for non-positive values in the effective ionization
coefficient. These negative values are obtained at low reduced electric fields depending on the mixture
composition (see the interrupted curves in figure II.10), and they suggest enhanced difficulties in the
breakdown of oxygen-containing plasmas.
Figure II.13: Electron-impact effective ionization coefficients, as function of E/N , for ground-state He
on the left and for excited He on the right, mixed with dry air (80% N2 - 20% O2 ).
35
The results on the left figure II.13 can be compared with those on the right side, in case of mixing
synthesized air with Helium with a relative density 10−4 of excited states, instead of pure ground-state
He. Since the concentration of metastables is higher than expected, it allows to clearly notice the effect of
stepwise ionization through these states on the effective ionization coefficient. Stepwise ionization clearly
contributes to increase the global coefficient at low reduced electric fields (below 30 Td, see section
II.3.1) and, therefore, compensates for the reduction caused by the attachment. In fact, we observe that
in presence of 10−4 He(23 S) and He(21 S), attachment only overcomes ionization in He-air mixtures in
particular regions of E/N and if the air concentration is as high as 50%. This fact reinforces the idea
that if the O2 concentration is low and the helium gas contains excited states densities, the attachment
process no longer consists of an impediment to the plasma development.
In figure II.14, on the left-hand side, we approach the analysis of He-air mixtures using pure groundstate He, this time noticing the evolution of the global ionization coefficient with air concentration for
the same three values of E/N considered before, neglecting attachment. The result is similar to those
for He-N2 mixtures and He-O2 mixtures, with maxima of Kion near 0.1% at 5 Td and near 1% at 20 Td.
Once again, it is the N2 and O2 direct ionization that increases with small amounts of air, as well as the
N2 and O2 excitation coefficients.
In the case of He-air mixtures without He metastables, equation II.1 for the power balance includes
all the power terms PE , Psup , Pel , Pexc and Pion , including the power gained by the electrons from
superelastic collisions, due to the N2 vibrational states. It means that all the energy gained by the electrons through the effect of the electric field and superelastic collisions is spent in elastic and inelastic
collisions and, in the latter case, this energy can be used in excitations or ionizations. Therefore, for
each calculated EEDF, and thus for each input value of mixture composition and reduced electric field,
a different amount of power is gained by the electrons and the energy losses will undergo a different distribution between the possible processes. On the right figure II.14, the fractional power lost in ionization
Pion /(Pel + Pexc + Pion ), is represented for the particular case of He-air (80% N2 - 20% O2 ) mixtures.
Figure II.14: Electron-impact ionization coefficients on the left and fraction of power transfered to ionization on the right, as function of the relative density of dry-air (80% N2 - 20% O2 ), for three values of
E/N : 5 Td, 20 Td and 60 Td.
As expected, the results show that the importance of ionization reactions is higher at high E/N (at
36
low air densities, Pion /Ploss is near 28% at 60 Td and near 7% at 20 Td) and very low at 5 Td. In this case
and coherently with the results of the ionization coefficients, we see a maximum of the power transfered
to ionization for E/N = 5 Td, when the concentration of dry air approaches 0.1%. Similar results are
obtained in He-O2 and He-N2 mixtures. If the He excited states are included, the power transfered
to ionization at 5 Td and at 20 Td slightly increases, in accordance with the results for the ionization
coefficients. The general trend of power loss distribution is that elastic collisions are more important at
low fields and at higher He densities, since He does not contain low energy states; vibrational excitations
gain importance for fields E/N between 0.1 Td and 100 Td and at higher N2 and O2 concentrations,
since these are the gases containing vibrational levels or for low fields if He excited states are included;
electronic excitations are particularly important at ∼ 30 Td in the case of pure He, ∼ 100 Td in the case
of pure O2 and ∼ 300 Td in the case of pure N2 ; ionization reactions always acquire significance with the
rise of E/N , being more significant for He than for O2 and for O2 than for N2 .
37
38
Chapter
III
Kinetic study of a pure Helium
atmospheric-pressure plasma
In order to develop collisional-radiative models (CRMs) for He-based plasmas, that can subsequently be
implemented in fluid models, and following the plan in section I.4, IST-LoKI needs to be developed to
include a CRM for He alone. As explained in section I.3.2, the study of the discharge kinetics requires
the solution of the rate balance equations (I.1), coupled with the electron Boltzmann equation (I.7). ISTLoKI works for imposed pressure p, gas temperature Tg , oscillation frequency f , tube radius R, electron
density ne and gas mixture with the corresponding electron-impact cross sections. The homogeneous
steady-state EBE is solved for the EEDF and the plasma swarm parameters and electron-impact rate
coefficients are calculated, as described in section II.1. Then, the set of rate balance equations, for
the different neutral and ion species, is solved using a Runge-Kutta method for ordinary differential
equations. The minimum time-step used is 1 ns. The reduced electric field E/N necessary to maintain
the discharge is self-consistently determined as an eigenvalue solution to the problem. The charged
particle net production rate must exactly compensate for the net loss rate, so as to satisfy the electron
ionization-loss balance equation, under the assumption of a quasi-neutral discharge. In order to find the
unique relationship between E/N and the product of the gas density and the discharge radius N R, E/N
is calculated. Since the discharge causes changes in the gas mixture composition, depending on each
E/N , the mixture for which the EEDF is calculated also requires self-consistency, under the demand
that the EBE is solved for the final E/N and gas mixture calculated by the chemistry solver. In the case
of pure He, the mixture for which the EBE is solved can also vary, due to the presence of the excited
states He(23 S) and He(21 S), as studied in chapter II. Finally, the rate balance equations are solved for
the same E/N and EEDF, for the whole temporal evolution, until steady-state.
In order to solve this system for He, we need to introduce new species, reactions and rate coefficients,
leading to the calculation of the corresponding steady-state densities and E/N . Then, the study can get
to conclusions about a scheme reduction for He CRMs and it can serve as a first step for further study
of He Patm post-discharges, pulsed-field systems and mixtures of He with N2 and O2 , which require the
implementation of other reactions and further adaptation of the chemistry solver. The zero-D simulations
39
of IST-LoKI do not intend to investigate the same results and phenomena as 2D or 3D simulations. By
considering only the application of a steady-state electric field and the chemical and diffusion processes,
we are isolating a part of the physics in the plasma from the global picture. The zero-D simulations allow
to understand the kinetics of the species and processes in the discharge, but care has to be taken before
extrapolating conclusions to different systems.
III.1
Helium collisional-radiative model and validation
In section I.2.2, it has been stated that in Santos et al. (2014) [12], an atmospheric-pressure CRM was
developed and validated, adopting a consistent set of electron cross sections and kinetic mechanisms. In
fact, this model is the reference from which a CRM for pure He is developed using IST-LoKI. Using the
reaction scheme in [12] as a starting point, a reduced scheme is first implemented in IST-LoKI, and its
validity is assessed. The reduced scheme is presented in table III.1. References [20–22] present similar
pure Helium schemes, although Wang et al. [22] emphasizes, in particular, the need for including the
atomic ion 3-body recombination (He+ + He + e− → He + He(23 S)), absent from the CRM in [12] and
in table III.1. In order to validate the numerical calculations from LoKI and the reduced scheme, the
same field oscillation frequency f = 2.45 GHz and tube inner-radius R = 3 mm as in [12] are used. In
[12], the gas temperature is self-consistently calculated through a thermal module not included in LoKI.
Therefore, the validation proceeds by imposing the same pairs of values of ne and Tg as those adopted
in [12].
The model in [12] has been partially described in section I.2.2 and the electron kinetic data needed
to calculate the EEDF has been presented in section II.2.1. The model adopts a kinetic scheme that
considers several electron and heavy species collision mechanisms involving electrons e− , He(n, l, s) ≡
3
+
He(n2s+1 l) excited states, ground-state atomic ions He+ , ground-state molecular ions He+
2 (2 SΣu ) and
He∗2 excimers. The CRM in [12] considers all the neutral atomic states up to level n = 7, while our
model only considers the densities of the most relevant neutral atomic states, with principal quantum
numbers n = 1 and n = 2, He(11 S), He(23 S), He(21 S), He(23 P ) and He(21 P ). The kinetic scheme in
[12] includes electron-impact excitation, de-excitation, direct ionization and stepwise ionization with cross
sections for electron energies up to 1 keV; recombination and electron-impact dissociation; associative
ionization mechanisms; charge transfer reactions between He+ and He+
2 ; internal energy transfer between
excited states of He and He2 ; radiative transitions between He excited states; transport of heavy particles
(He(23 S), He(21 S), He∗2 , He+ , He+
2 ). The coefficients of heavy particle reactions are either constant or
gas temperature dependent as shown in table III.1. As mentioned, notice that the scheme in table III.1
does not include the following mechanisms: excitation towards levels with n > 2, de-excitation from
these levels, stepwise ionization through levels different than He(23 S) and He(21 S), associative ionization
−
He(n > 2, l, s) + He → He+
and quantum number exchange reactions He(n > 2, l, s) + He →
2 + e
He(n > 2, l0 , s) + He.
The electron-impact rate coefficients for excitations (R1-9), de-excitations (R10-18) and ionization
(R19-21) are calculated through the energy integration over the EEDF (see eq. (I.8)) of the corresponding
40
Nr
R1-4
R5-7
R8-9
R10
R11-12
R13-15
R16-18
R19-21
R22-25
R26
R27
R28
R29
R30-39
R40-49
R50
R51
R52
R53
R54
R55
R56
R57
R58
Table III.1: Reaction scheme for He at atmospheric-pressure, reduced from [12].
Process
Reaction
Rate coefficient
e-impact exc.
He(11 S) + e− → He(2) + e−
EEDF (E/N )
He(23 S) + e− → He(21 S/23 P/21 P ) + e−
EEDF (E/N )
He(21 S) + e− → He(23 P/21 P ) + e−
EEDF (E/N )
e-impact de-exc.
He(23 S) + e− → He(11 S) + e−
EEDF (E/N )
He(21 S) + e− → He(23 S/11 S) + e−
EEDF (E/N )
He(23 P ) + e− → He(21 S/23 S/11 S) + e−
EEDF (E/N )
He(21 P ) + e− → He(21 S/23 S/11 S) + e−
EEDF (E/N )
e-impact ion.
He(11 S/23 S/21 S) + e− → He+ + 2 e−
EEDF (E/N )
−
1
Diss. recomb.
He+
EEDF (E/N )
2 + e → He(1 S) + He(2)
−
−10 0.71 −3.4/Te
Excimer ion.
He∗2 + e− → He+
+
2
e
9.75
×
10
Te e
cm3 s−1
2
+
−
3
−
−14 2.3
4.5
3-body recomb.
He + 2 e → He(2 S) + e
1.1 × 10 Tg /Te cm6 s−1
+
−
3
1
−
He2 + 2 e → He(2 S) + He(1 S) + e
0.5 × 1.1 × 10−14 Tg2.3 /Te4.5 cm6 s−1
+
−
∗
−
He2 + 2 e → He2 + e
0.5 × 1.1 × 10−14 Tg2.3 /Te4.5 cm6 s−1
−1.86
T (K)
cm3 s−1
Penning ion.
He(2) +He(2) → He+ + He(11 S) + e−
0.3 × 2.9 × 10−9 g300
−1.86
T (K)
−
He(2) +He(2) → He+
0.7 × 2.9 × 10−9 g300
cm3 s−1
2 + e
−0.6
T (K)
1
Charge transfer
He+ + 2 He(11 S) → He+
1.4 × 10−31 g300
cm6 s−1
2 + He(1 S)
Association
Dissociation
Radiative trans.
1
+
1
He+
2 + He(1 S) → He + 2 He(1 S)
3
1
∗
He(2 S) + 2 He(1 S) → He2 + He(11 S)
He(23 P ) + 2 He(11 S) → He∗2 + He(11 S)
He∗2 + He(11 S) → He(23 P ) + 2 He(11 S)
He∗2 + e− → 2 He(11 S) + e−
He(23 P ) → He(23 S) + hν
He(21 P ) → He(21 S) + hν
He(21 P ) → He(11 S) + hν
R59-60
Diff. to the wall
He(23 S/21 S)
R61
Diff. to the wall
He∗2
R62
Diffusion
He+
R63
Diffusion
He+
2
1.4×10−6 −28100/Tg (K)
cm3
Tg (K)−0.6 e
−34
6 −1
1.5 × 10
cm s
1.6 × 10−32 cm6 s−1
3.6 × 10−14 cm3 s−1
4.0 × 10−9 cm3 s−1
1.018 × 107 s−1
0.180 × 107 s−1
47.728 × 107 s−1
Tg (K)1.5 cm2
p(T orr)
s
Tg (K)1.5 cm2
D∗ = 7.102 × 10−2 p(T
orr)
s
2.68×1019
cm2
µa × N = 2.96×10−3
0.5
−2
Tg +3.11×10
Vs
k T (K) cm2
Da × N = µ × B qge
s
2
µm × N = 2.6 × 1019 cm
Vs
k T (K) cm2
Dm × N = µ × B qge
s
DM = 8.922 × 10−2
inelastic and superelastic cross sections, validated in section II.2.1. Several electron-ion recombination
mechanisms are considered. For the dissociative recombination of helium molecular ions (R22-25), we
use the collision cross section in the online supplementary data file associated to [12]. The corresponding
cross sections do not take part in the calculation of the EEDF but, instead, are integrated over the EEDF
through eq. (I.8), yielding the total dissociative recombination rate coefficient. The measurements of
[68] provide us branching ratios for the products issued from the dissociative recombination reaction:
1.1% He(23 S), 34.1% He(21 S), 64.5% He(23 P ) and 0.3% He(21 P ). The electron-stabilized recombination
of He+ (R27) is considered to produce only He(23 S) and to be temperature-dependent, as derived in
[65]. Similar reactions for He+
2 (R28-29), are included with rate coefficient and branching ratio br = 0.5,
adopted in [65].
This CRM considers Penning reactions (R30-49) between all pairs of metastable states He(23 S) and
He(21 S) and radiative states He(23 P ) and He(21 P ), with branching ratios 0.3 and 0.7 for the production
B
of He+ and He+
2 , respectively. The rate coefficient adopted follows the expression A(Tg (K)/300) , which
41
s−1
was fitted to the data in [20] at 300 K and in [65] at 2450 K. The ion conversion reactions (R50-51)
are expected to be important creation/destruction channels for ions in high-pressure regimes, due to the
high He density. The corresponding rate coefficients adopt the temperature dependence used in [20, 65].
Besides, the model includes several creation/destruction mechanisms of He∗2 excimers. The ionization
of He∗2 excimers by electron-impact (R26) is taken into account, adopting the rate coefficient proposed
by [69], featuring a dependence on the electron temperature, calculated by IST-LoKI using the EEDF.
The creation of He∗2 is assumed to occur from collisions of the He(23 S) and He(23 P ) states with neutral
ground-state atoms (R52-53), adopting the same rate coefficients as were used in [65]. The dissociation of
He∗2 by atomic impact (R54) is also expected, due to the low binding energy (1.7 eV) of this molecule, and
produces He(23 P ), with the rate coefficient adopted in [65]. The dissociation of He∗2 by electron-impact
(R55) also adopts the rate coefficient used in [65]. Radiative transitions (R56-58) are described using
the quantum default method to calculate the transition probabilities Aji = 4.333 × 107 (∆uji )2 (gj /gi )fji
s−1 , with fji the oscillator strengths 0.5374 for (R56), 0.3432 for (R57), 0.0734 for (R58) and 0 for the
other transitions from He(23 P ) and He(21 P ) to lower levels, due to the radiative transition selection rules
∆S = 0, ∆L = 1. Radiation imprisonment is not considered in this model, unlike in [12], where it is
stated that the excited-states populations are not high enough to cause radiation trapping but, on the
contrary, where the plasma is considered optically thick for transitions to the ground-state.
The transport by diffusion towards the walls is considered for metastable states with high mean
lifetimes (8400 s for He(23 S) and 2 × 10−2 s for He(21 S)), adopting the diffusion coefficient DM with
dependence on the gas density from [20] (R59-60). The diffusion of helium excimers He∗2 (R61) is also
taken into account, with a temperature and pressure dependence similar to that of the atomic metastables.
The mobility and diffusion coefficients for electrons are obtained from the calculated EEDF, through eq.
(II.2-II.3). The free diffusion coefficients of He+ , Da (R62), and of He+
2 , Dm (R63), are obtained from
the corresponding mobilities, using Einstein’s relation D/µ = kB Tg /qe , considering that the ions are in
thermal equilibrium with the atoms at temperature Tg . The reduced mobilities of the ions adopt the same
temperature dependence as in [12]. The loss through diffusion of the excited and ion species of He results
in the creation of ground-state He(11 S). The steady-state spatially averaged rate balance equation for a
given species j is written as
Sj '
Dj
nj
Λ2
(III.1)
where nj is the density of species j, Dj its diffusion coefficient, Sj its net production rate through
volume reactions and Λ ≡ R/2.405 the characteristic diffusion length, considering a cylindrical vessel.
At high-pressure, the electrons are considered in equilibrium with the average local electric field, and the
transport of charged particles is assumed to satisfy the ambipolar diffusion approximation. Therefore,
we assume the same rate of loss to the walls for electrons and for atomic and molecular ions, implying
that the ambipolar diffusion coefficients for charged species satisfy
na Da,a + nm Da,m ' ne Da,e
Da,e =
na (µa De + µe Da ) + nm (µm De + µe Dm )
na µa + nm µm + ne µe
42
(III.2)
(III.3)
nm (µm Da + µa Dm ) + ne (µa De + µe Da )
na µa + nm µm + ne µe
na (µa Dm + µm Da ) + ne (µm De + µe Dm )
=
na µa + nm µm + ne µe
Da,a =
(III.4)
Da,m
(III.5)
As charge conservation imposes Sa + Sm = Se , the production-loss equation for electrons can be obtained
from the corresponding equations for the ions.
The CRM in table III.1 is used in IST-LoKI and the results obtained are compared with those of
[12] for validation purposes. The calculations are for the same field oscillation frequency f = 2.45 GHz,
the same capillary tube radius R = 3 mm and the same pairs of values (ne , Tg ), where Tg is calculated
by the thermal module in [12]: (1.0 × 1013 cm−3 , 1534.5 K), (1.5 × 1013 cm−3 , 1734.6 K), (1.7 × 1013
cm−3 , 1800.6 K), (2.5 × 1013 cm−3 , 2031.8 K), (5.0 × 1013 cm−3 , 2507.0 K), (7.4 × 1013 cm−3 , 2797.9
K). However, these values were obtained in [12] taking electron-electron collisions into account, which
is not the case with IST-LoKI. Therefore, the code used in [12] was used to also obtain results without
electron-electron collisions for two pairs of values (ne , Tg ), (1.7 × 1013 cm−3 , 1826.6 K) and (7.4 × 1013
cm−3 , 2931.4 K). These are compared with the results in [12] and with the results from IST-LoKI.
The left figure III.1 depicts the self-consistent reduced electric field calculated with each code. In
addition to E/N , used along with the frequency as input to the EBE, the effective field Eef f /N is also
represented, defined as
Eef f
E
νc
=
N
N (νc2 + ω 2 )1/2
(III.6)
where ω = 2πf and νc is the total frequency of momentum-transfer electron-neutral collisions. Eef f /N
is used in the calculation of the EEDF and of the power terms. We observe in figure III.1 that the
differences between E/N and Eef f /N are small, which is expected at atmospheric-pressure when the
collision frequency is high, showing that the field oscillation frequency is not important in this study.
On the right-hand side of the same figure, the steady-state density of the ground-state species of He is
represented. This species consists on the majority of the gas and, therefore, its density is approximately
N = p/(kB Tg ).
Figure III.1: Reduced electric field on the left and He ground-state density on the right, as function of
the electron density, from [12] with and without electron-electron collisions and from IST-LoKI using the
reaction scheme from table III.1.
43
We notice in figure III.1 that the reduced electric field calculated in [12] is always lower than the
field calculated by IST-LoKI, between 0.5 and 1.5 Td (10% and 20%). The exclusion of electron-electron
collisions from [12] provides fields closer to those calculated by LoKI, but still ∼ 10% lower. These
results suggest that the CRM used by IST-LoKI is considering either less ionization or more chargedparticle losses than the CRM in [12], leading to an increase in E/N for compensation. Besides, E/N is
decreasing with ne in [12] but increasing in LoKI, which means that a more detailed analysis is required to
understand the differences between the two models. The results for the electron temperature show exactly
the same shape as those for the reduced electric field, since Te is calculated from the E/N -dependent
EEDF, with values between 1.793 eV and 1.890 eV in [12] and between 2.132 eV and 2.235 eV in ISTLoKI. Conversely, the comparison of the densities of the main species in the plasma, He(11 S), shows a
good agreement between the models, which was expected since the same conditions of gas pressure and
temperature are used, yielding values in the 2-5 × 1018 cm−3 . We notice that the density is slightly lower
using the CRM from [12] without electron-electron collisions, which is due to the higher calculated Tg .
We now focus on the charged-species. Since the electron density is fixed, the relevant analysis concerns
the distribution between atomic ions He+ and molecular ions He+
2 . In figure III.2, the relative densities
(nj /N ) of these ions are represented as function of ne for both models.
Figure III.2: Helium atomic ion relative densities on the left and molecular ion relative densities on the
right, as function of the electron density, from [12] with and without electron-electron collisions and from
IST-LoKI using the reaction scheme from table III.1.
We notice in figure III.2 that in [12] the atomic ion density is lower than the molecular ion density for
ne < 5 × 1013 cm−3 , whereas the results from LoKI show a He+ density higher than the He+
2 density, for
all values of electron density. This represents a big difference between the two models and, in fact, that
difference is reinforced by the fact that LoKI results show a He+
2 density that decreases with ne , while in
[12] this density is slightly increasing, and by the fact that LoKI presents always higher He+ and lower
+
He+
2 densities than [12]. In addition, the results for He present a difference between the two models of
up to two orders of magnitude at low ne , being closer at high ne , and the results for He+
2 show an even
higher difference, of up to 5 orders of magnitude at high ne and closer results at low ne . The results
obtained using the code of [12] without the electron-electron collisions do not show significant differences
by respect to those involving these collisions.
44
Even though there are differences in the EEDF calculations done in [12] and with LoKI, since our
model does not include electron-impact excitation cross sections towards levels He(n, l, s) with n > 4, we
have seen in section II.2.1 that the EEDFs provide similar swarm parameters using either LoKI or [12] in
this range of E/N ∼ 5 Td, which means that the cause for the differences presented in figure III.2 is not
the EEDF calculation. The results obtained from IST-LoKI with the reaction scheme from table III.1
and their differences with respect to [12] are explained by changes in the creation and loss mechanisms
of the species considered.
In figure III.3, important charge creation rates are represented. On the left side it is depicted the
sum of the rates of all associative ionization reactions from He(n, l, s) with n > 2 from [12], which are
not included in the CRM of table III.1. On the right side, the most important charge creation rate in
the LoKI CRM is represented, the stepwise ionization reaction. In IST-LoKI, this reaction includes only
the He metastables He(23 S) and He(21 S), while in [12] electrons collide with all the He excited states to
perform stepwise ionization.
Figure III.3: Creation rates of electrons and ions through associative ionization (left) and through stepwise
ionization (right), normalized to the total creation rates of charged species, as function of the electron
density, from [12] with and without electron-electron collisions and from IST-LoKI using the reaction
scheme from table III.1.
The exclusion of species He(n > 2, l, s) from the CRM of table III.1 implies the exclusion of electronimpact excitation towards those species, of superelastic collisions with those species and of radiative
transitions from those species. But it also excludes the associative ionization reactions and the quantum
number exchange reactions, that also involve the He(n > 2, l, s) species. In fact, we observe in figure III.3
that the associative ionization reactions are highly influent in electron and He+
2 creation, with relative
rates between 77% and 88% of the total electron creation rates. The other electron creation rates, the
direct electron-impact ionization (R19), the stepwise ionization (R20-21), the Penning ionization (R3049) and the excimer ionization (R26), of which the stepwise ionization has the highest rate in [12],
contributing between 11% and 22% to charge creation. Therefore, the lack of associative ionization in the
CRM of table III.1 leads to a higher importance of stepwise ionization, with rates between 1 × 1017 and
7 × 1017 cm−3 s−1 and relative contribution to charge creation between 96% and 99%. The exclusion of
associative ionization explains also why [He+
2 ] is lower in LoKI than in [12] and the enhanced influence
45
of stepwise ionization justifies the higher [He+ ] in LoKI than in [12] (see figure III.2). The difference is
lower for He+ at high ne , since then associative ionization loses importance and He+ acquires relevance,
being mostly created and destroyed by charge-transfer reactions (R50-51).
The exclusion of species He(n > 2, l, s) from the CRM of table III.1 was done in order to try a kinetic
scheme more reduced than the one in [12]. The inclusion of a single state He(n > 2) was considered as
possible upgrade. This state would be created by the sum of the electron-impact excitations towards
all the states with n > 2 and destroyed by the associative ionization reaction. However, we would have
to consider a single associative ionization rate coefficient as a weighted average of all the associative
ionization rate coefficients, which would deform the model by implicitly assuming the same coefficient for
all the excited states. Detail would be lost but the global effect on the ionization mechanisms could be
achieved. The inclusion of other particular He excited species alone was also considered. However, in the
studied conditions, there is no n > 2 state with relative density higher than 10−8 and, therefore, there
is no particular species alone that would make a big difference. Moreover, the rate coefficients presented
for associative ionization in [12] are relative to a gas temperature of 2450 K and obtained from [65].
Therefore, the inclusion of these reactions should consider a gas temperature-dependence formulation of
the rate coefficients.
In figure III.4, the same type of comparisons of figure III.3 are made but now for the most relevant
charged-species loss rates, dissociative recombination (R22-25) on the left and ion/electron diffusion
(R62-63) on the right. The relative rates are calculated over the total loss rate of the charged species.
Figure III.4: Loss rates of electrons and ions through dissociative recombination (left) and through
diffusion (right), normalized to the total loss rates of charged species, as function of the electron density,
from [12] with and without electron-electron collisions and from IST-LoKI using the reaction scheme from
table III.1.
The dissociative recombination rates on the left figure III.4 are proportional to the He+
2 densities,
which is why these rates decrease when calculated using LoKI, as do the densities of He+
2 in figure III.2.
In [12] this is the dominant charge loss reaction, with relative rate between 97.3% and 99.1%, as expected
in high-pressure plasmas. In LoKI, as the dissociative recombination loses relevance (from 64.6% at low
ne to 0.1% at high ne ), the diffusion of charged particles replaces it as the dominant charge loss reaction,
mainly due to He+ , with relative rates between 35.4% and 99.2%. The electron-stabilized recombinations
46
of He+ and He+
2 (R27-29) have relative rates below 1% in both cases. Notice the influence of the charge
exchange reactions (R50-51) in the ion balance. In [12] these reactions are the dominant mechanism
for the destruction of He+ for all ne , for the creation of He+ at high ne and for both the creation and
destruction of He+
2 at high ne , since the ion densities tend to increase with ne to fulfill the quasi-neutrality
requirement. Using the CRM of table III.1, these reactions have relevance for the balance of ion densities
for every condition. They contribute as much to the creation of the ions as they do to their destruction,
with relative rates always over 95%.
Regarding the Helium atomic and molecular excited species, their relative densities in both models
are compared. The metastables He(23 S) and He(21 S) that take part in the EEDF calculation are shown
in figure III.5. This figure compares results obtained with the two models, [12] and LoKI. The same kind
of comparison is presented in figure III.6 for two other relevant excited species, He(23 P ) and He∗2 .
Figure III.5: Helium atomic metastable densities He(23 S) on the left and He(21 S) on the right, normalized
to the total gas density, as function of the electron density, from [12] with and without electron-electron
collisions and from IST-LoKI using the reaction scheme from table III.1.
Figure III.6: Helium densities of atomic radiative species He(23 P ) on the left and molecular excimer He∗2
on the right, normalized to the total gas density, as function of the electron density, from [12] with and
without electron-electron collisions and from IST-LoKI using the reaction scheme from table III.1.
In general, the differences in these densities calculated using the two models are less than one order of
magnitude. The models agree that the relative densities of He(23 S), He(23 P ) and He∗2 are close to 10−7
47
(in absolute values ∼ 1011 -1012 cm−3 ), while the densities of He(21 S) and He(21 P ) are close to 10−8 (∼
109 -1011 cm−3 ). However, in LoKI the density of He(21 S) is higher than that of He(21 P ) for every ne ,
while in [12] the opposite happens. Furthermore, we notice that there is a significant decrease in the
density of He∗2 in [12], getting close to [He(21 S)], while in our model these densities keep one order of
magnitude apart.
In both models, He(23 S) is destroyed by electron-impact excitation and stepwise ionization. The
He(23 S) creation processes are also common. At low ne , radiative transition (R56) (∼60% of the He(23 S)
creation rates), electron-impact de-excitation from He(23 P ) (R14) (∼30%) and from He(21 S) (R11)
(∼5%) are the relevant processes. At high ne , the same processes dominate but with different importances:
electron-impact de-excitation can contribute up to ∼80% and radiative transition can decrease down to
∼20% of the creation rates. He(23 P ) is created by both electron-impact excitation from He(23 S) (R6)
3
and by He+
2 dissociative recombination (R24). However, (R6) contributes only with ∼2% for the He(2 P )
creation in LoKI, while in [12] this contribution goes up to ∼ 20%, due to the difference in [He+
2 ]. The
He(23 P ) destruction processes are the same in both models. These atoms are destroyed at low ne by
radiative transition (R56) (∼65%), by electron-impact de-excitation towards He(23 S) (R14) (∼30%) and
by creation of He∗2 (R53) (∼2%). At high ne , the electron-impact process (R14) acquires importance up
to ∼80% and radiative transition (R56) decreases contribution down to ∼20%. He∗2 is mostly created by
the association reaction (R53) (95-99% importance) in both models, but in [12] the recombination (R29)
acquires an importance of ∼ 5%, which does not happen in our model, due to the low [He+
2 ]. The excimer
is destroyed by the same processes at low ne in both models: electron-impact dissociation (R55) (∼20%)
and atomic impact dissociation (R54) (∼80%). At high ne , as the electron-impact processes acquire
importance, He∗2 is destroyed by electron-impact dissociation (∼70%), by atomic impact dissociation
(∼23%) and also by electron-impact ionization (R26) (∼6%), as emphasized in [22].
The reaction rates for the creation and destruction of the least relevant species He(21 S) and He(21 P )
are also similar in both models. At low ne , He(21 S) is created by excitation from the ground-state
(R2) (∼20%) and from He(23 S) (R5) (∼60%) and by de-excitation from He(23 P ) (R13) (∼13%) and is
destroyed by electron-impact excitation towards He(23 P ) (R8) (∼5%) and towards He(21 P ) (R9) (∼90%)
and by electron-impact de-excitation towards He(23 S) (R17) (∼5%). At high ne , the rates for He(21 S)
creation and destruction slightly change, but the dominant processes remain the same. Finally, He(21 P )
is created at low ne by electron-impact excitation from He(11 S) (R4) (∼15%), from He(23 S) (R7) (∼15%)
and from He(21 S) (R9) (∼65%) and destroyed mostly by the radiative transition towards the groundstate (R58) (∼99%). At high ne , the only significant change in He(21 P ) rates, is the increase in the rate
of electron-impact de-excitation towards He(21 S) (R16) up to ∼ 3%.
We can conclude that the reactions in [12] missing in table III.1, which are associated to the atomic
states He(n > 2, l, s), do not have a significant influence in the rates and densities of the He atomic and
molecular excited species. However, they strongly affect the ionization mechanisms and the plasma ionic
composition. The plasma kinetics is highly influenced by the missing associative ionization involving
He(n > 2, l, s) states, but not by the other He(n > 2, l, s) reactions. Electron-impact reactions with
these states, radiative transitions considering these states and quantum-number exchange reactions would
48
slightly change the densities of each particular excited species, but not the global dynamics of the plasma.
The orbital quantum number exchange reactions may be important in establishing a local statistical
equilibrium between the different angular momentum sublevels of a particular principal quantum number
level n, but that is not important in establishing a reduced scheme to study the plasma dynamics. The
CRM used in IST-LoKI, with the scheme from table III.1, is partially validated through this comparison
with [12] and will be used for the study of discharge kinetics in conditions different than those of [12].
The present analysis has allowed to conclude that a reduced kinetic scheme for Helium needs to include
the He excited species He(n > 2) and the Penning and associative ionization reactions and, therefore,
has given an indication for the future development of the He CRM using IST-LoKI. The results from this
comparison have also given insight on the importance of the densities and the creation and destruction
reaction rates of each species, which will be studied with further detail in the next section.
III.2
Atmospheric-pressure discharge results
The study of Helium-based CRMs gets us closer to study the plasma in the conditions of the jets presented
in section I.3.1. Therefore, after having shown the validity and limitations of the a Helium reduced CRM
in the previous section, the tool IST-LoKI is used to study a steady-state plasma maintained by a constant
applied electric-field, but in closer conditions to the plasma jets. As in [26], the gas temperature is set
to the room temperature Tg = 300 K, the pressure remains atmospheric p = 1.013 × 105 Pa and the
tube radius is set equal to the one used in the experiments in GREMI, R = 2 mm [23, 27, 28]. In these
conditions of Tg and p, the total gas density remains constant at N = 2.45 × 1019 cm−3 . The field
is taken as DC, with f = 0, and the imposed electron density is varied to cover the range obtained in
[23, 26, 28], with values 109 , 1010 , 1011 , 5×1011 , 1012 , 5×1012 , 1013 , 5×1013 and 1014 cm−3 . The numerical
parameters for the resolution of the EBE are chosen in order to provide enough physical detail, good
electron power balance and short software run-times: uM AX = 50 eV, N = 2500 points, ∆u = 0.02 eV.
Figure III.7: Reduced electric field E/N on the left and electron temperature Te on the right, as function
of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1.
On the left figure III.7, the reduced electric field E/N is represented for the described conditions. E/N
is calculated for the final mixture, iterating over the quasi-neutrality condition as to satisfy the criterion
49
[He+ ]+[He+
2 ]−ne
ne
≤ 10−8 . The same figure depicts Te =
2
3
< u > on the right-hand side, calculated through
the EEDF(E/N ) corresponding to each ne . As expected, the imposition of a higher electron density leads
to a higher field, which also increases Te , whose values vary between ∼ 1.6 eV (∼ 18 000 K) and ∼ 2.7
eV (∼ 30 000 K). The values of E/N calculated for ne = 1013 cm−3 and ne = 5 × 1013 cm−3 , 5.5 Td and
6.1 Td, respectively, are slightly higher than those studied in figure III.1 using the same tool IST-LoKI,
for different conditions of R and Tg . This difference in the plasma conditions leads to different rates of
the reactions in table III.1, which will be studied with more detail later in this section. We know that
the diffusion rates depend on R, according to eq. III.1, which should not have much influence at high
pressures. We see also in table III.1 that some rate coefficients (R26-51) and (R59-63) explicitly depend
on Tg . But most importantly, the difference in Tg leads to a different gas density N and, therefore, to
a different density [He(11 S)] considered in the calculation of the reaction rates. The ensemble of these
factors leads to different values of E/N presented in figures III.7 and III.1. Since we know that these
values are in the 4-7 Td range, we can deduce from chapter II that we are working in a region where the
densities of metastables have a very influent role on the EEDF, on the swarm parameters (including the
excitation and ionization coefficients) and on the power transfer terms (see figures II.6, II.7, II.8, II.13,
A.7, A.8 and A.9).
For the particular case of ne = 1013 cm−3 and constant E/N = 5.5 Td, we observe in figure III.8
how the species densities evolve in time. The species rate equations are solved from t = 10−9 s until t =
10 s, when the steady-state has already been achieved. Even though there is a temporal evolution of the
ion densities, ne is always taken with the same value, which means that the quasi-neutrality condition is
only attained at equilibrium, in the case of figure III.8 near t = 100 µs.
3
1
Figure III.8: Helium ion species He+ and He+
2 densities (left) and excited species He(2 S), He(2 S),
He(23 P ), He(21 P ) and He∗2 densities (right), as function of time, calculated by IST-LoKI using the
reaction scheme from table III.1, at ne = 1013 cm−3 and E/N = 5.5 Td.
The species densities evolve differently with time and, therefore, so do the reaction rates. However,
since the applied electric field is constant and we are looking for the characteristics of the discharge
in equilibrium, hereinafter the densities and rates will be taken at equilibrium, when the rate-balance
equations for every species j have achieved
∂nj
∂t
= 0.
In figure III.9 we observe the results for the densities of every He charged and excited species in the
50
plasma, as function of ne and, therefore, for different values of E/N and Te according to figure III.7. The
electron density is also explicitly represented, for reference.
−
Figure III.9: Charged species He+ , He+
densities (left), and excited species He(23 S), He(21 S),
2 and e
3
1
∗
He(2 P ), He(2 P ) and He2 densities, along with e− density (right), as function of the electron density,
calculated by IST-LoKI using the reaction scheme from table III.1.
The results on the left figure III.9 show that for every value of electron density He+
2 is the dominant
ion and the quasi-neutrality condition is obtained as ne ' [He+
2 ]. This is very different from the results
studied in the previous section (figure III.2), where IST-LoKI yielded He+ as the dominant ion for every
ne . On the right figure III.9, we observe that the excited species densities also increase with ne and
E/N , which is primarily due to the fact that electron-impact reaction rates are proportional to ne and to
rate coefficients that increase with E/N , due to the more energetic EEDF. We can conclude from figure
III.9 that the relation between the excited species densities is variable with ne and that the metastable
He(23 S), the radiative state He(23 P ) and the excimer He∗2 can have important densities, overcoming
ne in some occasions. He(21 P ) always presents rather low densities and a more reduced scheme might
consider its removal. In the previous section, the relative electron/ion densities were ∼ 10−5 − 10−6
(figure III.2), the densities of He(23 S), He(23 P ) and He∗2 were approximately 1 order of magnitude lower
and the densities of He(21 S) and He(21 P ) were about 10 times lower than the latter (figures III.5, III.6).
In the present section the relation between the orders of magnitude of these species is not the same, even
in the ne = 1013 − 1014 cm−3 range, where [He∗2 ] surpasses ne . The relative densities of the metastables
He(23 S) and He(21 S) are variable between ∼ 10−9 when ne = 109 cm−3 and ∼ 10−6 if ne = 1014 cm−3 ,
which is why, according to section II.3.1, they can have some influence on the EEDF and on all the
parameters calculated from the EEDF. The explanation for the values of the species densities can be
given by studying the reactions responsible for each species balance.
Figure III.10 represents the creation/destruction rates of He+
2 , calculated for the different ne .
12
14
Excimer ionization (R26) appears to be important in He+
2 creation, specially at high ne = 10 − 10
cm−3 , since its rate increases with Te and with ne . Charge-transfer reaction (R50) also proves to be
essential for the balance between the ion densities, for every ne . The whole set of reactions (R40-49)
refers to Penning ionization reactions. We notice that the sum of their rates is very important and
that some reactions should be highlighted as follows. Reactions (R40-42), that depend on [He(23 S)],
51
Figure III.10: Rates of creation (left) and destruction (right) of He+
2 , discriminated by reaction, as
function of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. The
numbers of reactions refer to table III.1.
[He(21 S)] and [He(23 P )] are relevant at low ne and reactions (R40,42,47), that depend on [He(23 S)] and
[He(23 P )] are relevant at high ne , while reactions (R43,46,48,49), that depend on [He(21 P )] appear to
have irrelevant rates, which reinforces the possibility of removing He(21 P ) from a more reduced model.
13
Comparing with the He+
cm−3 , where Tg = 1536
2 creation rates of the previous section for ne = 10
K, we acknowledge that the charge-transfer (R50) rate is 2 orders of magnitude higher due to the higher
[He+ ] but now, at 300 K gas temperature, the excimer ionization (R26) rate is 20 times higher and the
Penning ionization (R40-49) rates are 2 to 3 orders of magnitude higher, due to the lower Tg and the
higher excited species densities.
Analyzing the destruction rates, we see that diffusion (R63) is important at low ne and that dissociative
recombination (R22-25) is essential at high ne , as described in [12]. On the contrary, electron-stabilized
recombination (R28-29) always has low influence, suggesting that these reactions can be discarded from a
reduced scheme. Charge transfer towards He+ (R51) is surprisingly low, not even observable in the figure,
due to the low Tg . By comparing these results with the ones of section III.1, even though dissociative
recombination and ambipolar diffusion rates are higher when Tg = 300 K mostly due to the higher [He+
2]
and E/N , there is a huge difference in the charge-transfer rate (∼ 30 orders of magnitude lower), this
being the main factor explaining the difference in the densities of the ion species. In fact, in the current
case the charge transfer reactions (R50-51) only happen in one direction, transforming He+ into He+
2
and not the other way around. In the future, it can be interesting to study the balance between the two
coefficients in detail for several conditions.
+
Although He+
2 is the dominant ion and He has low density, the rates of creation and destruction of
this ion still have relevant values and can influence the charge balance. These rates are represented in
figure III.11.
We learn from figure III.11 that electron-impact ionization reactions (R19-21) are very important for
+
He , even at low E/N between 4 and 7 Td. Stepwise ionization rates are particularly important, even
with relative densities of He(23 S) and He(21 S) varying between 10−9 and 10−6 , confirming the analysis
of section II.3.1 for the global ionization coefficients. As mentioned earlier, charge transfer from He+
2
52
Figure III.11: Rates of creation (left) and destruction (right) of He+ , discriminated by reaction, as
function of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. The
numbers of reactions refer to table III.1.
towards He+ (R51) is so low that it is not observable in these figures. The analysis of the set of Penning
ionization reactions (R30-39) is the same as earlier for (R40-49), highlighting the role of He(23 S) and
He(23 P ) in rates (R30,31,32,37) and reinforcing the irrelevance of He(21 P ) in rates (R33,36,38,39). Once
again, comparing the He+ creation rates in both models (Tg = 300 K and 1536 K) evidences that chargecreation rates (R19-21,R30-39) are higher in the model presented in this section, due to the higher E/N ,
the higher excited species densities and the lower Tg , but the charge-transfer rate (R51) is ultimately the
responsible for the decrease in [He+ ].
Regarding the destruction of He+ , the results are as expected. Electron-stabilized recombination
(R27) has an even lower rate than in the previous section, due to the lower Tg and the lower [He+ ], and
can be considered for removal from a reduced scheme. The diffusion rate (R62) presents also low values
(3 orders of magnitude lower than in section III.1), for the same reasons, but should not be discarded,
since it can be important at slightly different conditions. Even though the charge-transfer rate towards
+
He+
2 (R50) is lower than in the previous section by 2 orders of magnitude, due to the lower [He ], it is
the dominant reaction leading to the disappearance of He+ .
Discarding charge transfers, if we consider the rates of charge creation and destruction only, we can
+
find the synthesis of the analysis made for He+
2 and He in figure III.12, where the set of reactions that
have influence on charge balance is represented. The reaction rates are assembled by groups.
The results in figure III.12 confirm the previous analysis for the ion rates. Penning ionization (R30-49)
and stepwise ionization (R20-21) rates are always very important for charge creation in the discharge,
reinforcing the role of the internal energy contained in the He excited states, transfered to ionization.
Excimer ionization proves to be relevant at high ne and Te . Dissociative recombination (R22-25) and
diffusion (R62-63) are essential for charge losses, while the electron-stabilized recombination rates are
negligible.
The He excited species (see figure III.9 representing their densities), that have shown their importance
for the plasma ionization, should also have their creation and destruction rates analyzed. Figure III.13
depicts these rates for the He(23 S) balance.
53
Figure III.12: Rates of creation (left) and destruction (right) of electrons, discriminated by reaction, as
function of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. The
numbers of reactions refer to table III.1.
Figure III.13: Rates of creation (left) and destruction (right) of He(23 S), discriminated by reaction, as
function of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. The
numbers of reactions refer to table III.1.
Analyzing the left figure III.13, we conclude that the electron-impact excitation rate (R1) is essential
as primary excitation of He and its value is higher at Tg = 300 K than at Tg = 1536 K, due to the
higher E/N and, most importantly, due to the higher [He(11 S)]. The electron-impact de-excitation
from He(21 S) (R11) is not irrelevant for He(23 S) creation and actually it is essential for the equilibrium
between the He excited-states densities. The transitions to He(23 S) from He(23 P ) through both radiation
(R56) and electron-impact de-excitation (R14) are important for both these species, particularly at high
ne and E/N , in which case He(23 P ) has a more relevant role. The de-excitation from He(21 P ) (R17) is
not relevant for He(23 S) creation, even at high ne , which contributes to reinforce the idea that a reduced
3
scheme can dismiss He(21 P ). The dissociative recombination of He+
2 , that creates He(2 S) (R22), is
not important for its creation but maintains its relevance for charge loss. Finally, electron-stabilized
recombination reactions that produce He(23 S) (R27-28) are again irrelevant for the species balances.
From the right figure III.13, we notice that stepwise excitation reactions from He(23 S) to higher
states (R5-7), and towards He(23 P ) (R6) in particular, have important rates for the He(23 S) destruction,
specially at high ne . In fact, these rates are approximately three times higher at Tg = 300 K than at
54
Tg = 1536 K, when ne = 1013 cm−3 , due to the He(23 S) density, which also triples. Stepwise ionization
(R20) is also relevant at high ne and the association reaction to form He∗2 (R52), whose rate shows almost
no variation with ne due to the constant coefficient and He(11 S) density, assumes an important role in
He(23 S) loss at low ne , below 1013 cm−3 . Even though the diffusion of He(23 S) (R59) is the least relevant
reaction for the species destruction for ne > 1011 cm−3 , it is still worth including at lower ne and in case
the diffusion acquires importance in a tube with lower radius. The superelastic electron-impact reaction
of He(23 S) towards He(11 S) (R10) and the Penning ionization reactions with He(23 S) (R30-33,40-43)
are not important for this species destruction but they remain relevant for the scheme consistency and
the discharge ionization.
The excimer He∗2 was shown to have a very important density in figure III.9, overcoming the densities
of He(23 S) and electrons for ne > 1012 cm−3 . It was also shown to have an important role in ionization
through (R26) in figure III.12. The rates of production and destruction of this species are represented in
figure III.14.
Figure III.14: Rates of creation (left) and destruction (right) of He∗2 , discriminated by reaction, as function
of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. The numbers
of reactions refer to table III.1.
As we see on the left figure III.14, the electron-stabilized recombination (R29) provides a negligible
rate for He∗2 production, about 7 to 9 orders of magnitude lower than the dominant reaction rates. On
the other hand, the association reactions through He(23 S) and He(23 P ) (R52-53) have total influence on
the He∗2 creation. The association with He(23 S) (R52) is dominant at low ne and relevant at high ne ,
while the reaction with He(23 P ) (R53) is dominant in the whole range of ne . At ne = 1013 cm−3 , these
rates are about 100 times higher at Tg = 300 K than at Tg = 1536 K. Since the reaction rate coefficient
is the same, this is due to the He(11 S) density that is 5 times higher and to the excited states densities,
that are approximately 3 times higher. The importance of He∗2 for the discharge is well shown in the
right figure III.14. The diffusion of this species (R61) is negligible for its balance but the dissociation
reactions (R54-55) define how He∗2 is destroyed and how it yields back He(11 S) and He(23 P ). The main
interest of the excimer is its role in ionization through the electron-impact reaction (R26). In our case,
the rates of (R26) and (R55) are about 20 times higher than at Tg = 1536 K and the rate of (R54) is ∼
2 orders of magnitude higher if Tg = 300 K. These differences are due to the densities of He∗2 , 20 times
55
higher and He(11 S), 5 times higher when Tg = 300 K.
The rates describing the balances of the other He atomic excited species He(21 S), He(23 P ) and
He(21 P ) are not represented here. He(21 S) is important for stepwise (R21) and Penning ionization
(R34-36,44-46), represented in figure III.11. Its population is essentially determined by electron-impact
reactions, redistributing the populations of the excited He(2) levels. He(23 P ) also has influence in this
redistribution, including through the radiative transition (R56). However, it also plays a role in the
densities of the excimer He∗2 through reactions (R53-54) and it has a big relevance in Penning ionization
rates (R37-38,47-48). Finally, the only role of He(21 P ) are the density exchanges with He(11 S) and
He(21 S) through electron-impact reactions and through radiative transitions (R57-58).
Following the analysis made in this chapter, we conclude that the He CRM requires the inclusion of
the excited species He(n > 2, l, s) and, in particular, the consequential associative ionization reactions.
However, we have been able to extract a lot of information on the discharge kinetics using the scheme of
table III.1 in conditions close to the jets presented in section I.3.1 and using a steady-state model with a
constant applied E/N . Thinking about defining a more reduced scheme, we have learned that the species
He(21 P ) has a negligible role and thus it can probably be disregarded, along with its reactions, including
Penning ionization (R33,36,38,39,43,46,48,49). The electron-stabilized recombination reactions (R27-29)
+
can also be discarded due to their small influence. The charge-transfer rate from He+
2 towards He (R51)
has proved to be highly sensitive to Tg and, therefore, this reaction can only be excluded if the model
imposes a low Tg , such as 300 K. All the other reactions in the set of table III.1 are recommended to
remain part of a reduced scheme for an atmospheric-pressure and room-temperature plasma, with He as
the majoritary gas. The extrapolation of these conclusions to cases using either a more complete scheme
with He(n > 2, l, s) species or a jet with a pulsed electric field, has to be done carefully.
56
Chapter
IV
Zero-dimensional simulation of an
atmospheric-pressure He-N2 tube streamer
IV.1
Formulation and tools
Following the previous chapters where we have studied the electron kinetics and related parameters in
He-based mixtures and the kinetics of a plasma with Helium gas alone at steady-state, we now want to
move forward to the study of a He plasma with impurities of the air gases N2 and O2 . In fact, in the
EM2C laboratory of ECP, along with a team of researchers, I have studied the zero-dimensional kinetics
of a He-N2 plasma, simulating the conditions of an ionization wave in a tube at atmospheric-pressure
described in section I.3.1. According to the plan in section I.4, understanding the influence of small
concentrations of N2 on the characteristics of a He-based discharge propagating along a dielectric tube
is set as the objective. In order to couple zero-D results for kinetics alone with 2D results of discharges
dynamics in the tube, the variation of N2 concentration needs to be introduced to the 2D discharge code.
In the EM2C, the electron Boltzmann equation solver BOLSIG+ [17], alongside the database website
LXCat [52], was used to easily change the N2 concentration in He-N2 mixtures and obtain the corresponding transport parameters and reaction rate coefficients. The website LXCat furnishes several cross-section
databases from several contributors. Three databases were compared for both He and N2 : Morgan [55],
Biagi [70] and IST-Lisbon [66]. The IST-Lisbon database was chosen. Using the cross-sections as input,
the off-line version of BOLSIG+ gives us tables of the wanted parameters for the required reduced electric
field values. In this case, a total of 194 E/N values were used between 10−5 and 3900 Td. The default
configurations of BOLSIG+ off-line were used. The parameters obtained from these E/N values can then
be used in the model. To obtain the parameters for other E/N values, which is something the 2D simulation model will be doing for every time-step and every point in space, linear interpolations were used to
write a subroutine to implement in the 2D discharge code. This method takes little calculation time and
it was confirmed that the adjustment is satisfactory using the 194 points. Other interpolation methods
were tried, like logarithmic ones, but the calculation time was higher and there was no improvement in
quality. This was done for several values of N2 concentration in the mixture and several parameters po57
tentially used in the model: electron mean energy, electron mobility, electron diffusion coefficient, energy
mobility, energy diffusion coefficient, transported power density and the electron-impact rate coefficients
that take part in the model. The rate coefficients for He-N2 mixtures have been studied in further detail
in section II.3.2.
In order to study the zero-D estimation of the ionization wave plasma in the tube, an on-line free-ware
zero-dimensional plasma kinetics solver was used to study the temporal evolution of He-N2 discharges.
ZDPlasKin [59] is a Fortran 90 module designed to follow the time evolution of the species densities and
the temperatures in a non-thermal plasma with an arbitrary complex chemistry. An input data file in an
easy-to-read format defines the species, the chemical reactions and their rate coefficients and the plasma
conditions. The rate coefficients may be constant or functions of the gas temperature or other plasma
conditions, as specified by the user. Only electron impact reactions, chemical reactions and light emission
reactions were written as input. Other processes such as diffusion losses, interaction with surface charges
or photo-ionization were not considered in this zero-dimensional model. A preprocessor converts the input
text file into a customized Fortran module with the input data for the plasma chemistry incorporated
directly into the code. The user must provide a master code to call the ZDPlasKin library routines
that perform the desired calculations. Unlike IST-LoKI, used in the previous sections, here the applied
reduced electric field at each time can be inserted through the master code. All the species densities are
calculated as function of this field at each time, including the electron density. In this work, the choice
for the electric field is an estimation of the field in the discharge front from the 2D simulation, as the one
in figure I.7. It has first been used a peak with a minimum of 0.4 Td and a maximum of 50 Td, centered
around 10 ns, with a rise time of 8 ns and a width = 0.2 ns. The field is shifted by shif t = 6 ns. The
slope rate is λ = 8/trise . Its formulation is given by:
E = Emin + (Emax − Emin ) × (
1
1+
e−λ×(t−shif t)
+
1
1+
e+λ×(t−shif t−width−trise )
− 1)
(IV.1)
Using this model, the region of time where the field is applied is a simulation of the discharge front
and the time posterior to the field application, which we call post-discharge, simulates what happens in
the channel after the passage of the ionization front. In the figures presented from here forward, such
as figure IV.1, the electric field is represented in the right-side inverted y-scale. Pre-ionization is defined
in the master code as 109 cm−3 for e− and N+
2 , the species that stay longer in post-discharge. This
level is taken into account to simulate the ionization left by previous discharges and will be discussed in
section IV.2.2. The gas temperature is kept constant at 300 K and the electron temperature is verified to
change by the effect of the electric field, presenting a peak at the same time with a maximum of electron
temperature around Te = 75 000 K (6.5 eV) for the studied cases.
For each instant in time, ZDPlasKin finds the user-defined local electric field and automatically calls
the electron Boltzmann equation solver BOLSIG+, that provides a rapid calculation of electron transport
parameters and reaction rate coefficients. The user must furnish an input data file with the cross-sections
that BOLSIG+ needs to use. Rate equations (I.1), that may use the rate coefficients from BOLSIG+
and the gas temperature, are integrated in time using the DVODE.F90 solver [59]. Default values of the
58
internal model parameters (ex. time-steps, error tolerances, etc) can be modified by the user. During the
work shown in this chapter, the time-step was set to 10−11 s. Diagnostics from ZDPlasKin are called from
the master code and provide a convenient way of visualizing the evolution of the different contributions to
the creation and loss of each species. This zero-D simulation does not intend to present the same results
and phenomena as the 2D simulation. By considering only the application of an ionization front electric
field and the chemical processes, we are isolating a part of the physics in the discharge propagation from
the global picture. The zero-D simulations allow to understand the kinetics in the plasma but care has
to be taken before extrapolating conclusions to more complex systems.
IV.2
Reaction schemes, results and discussion
In order to understand the influence of the N2 density in the discharge dynamics, we need to understand
the kinetics in the He plasmas with N2 impurities. And first of all, we have to understand the kinetics in
the original model, described in section I.3.1 and in [26] with 1000 ppm of N2 and using the reaction scheme
of table B.1 in appendix B. The kinetic scheme used is derived mostly from [29] and from [30]. It intends
to represent a reduced scheme of He-N2 reactions for low reduced fields, E/N < 80 Td. Three positive
+
3
1
ions are considered in this scheme (He+ , He+
2 , N2 ), as well as three excited species (He(2 S)+He(2 S),
N2 (C), N+
2 (B)). Reactions (R15) to (R18) are taken into account to compute the time evolution of the
N2 (C 3 Πu ) and derive the emission of the N2 second positive system N2 (C → B), which can be compared
with optical emission spectroscopy experiments [23, 71]. The local field approximation is used and the
transport parameters and electron-impact reaction rates of the model are functions of the local reduced
field E/N (called E in this chapter). The rate coefficients (R1, R2, R3 and R15) are taken from Bolsig+
[17], calculated from the IST-Lisbon cross-section database from LXCat [52, 66] for He and N2 . As far
as the reactions for He alone are concerned, we notice that the scheme in table B.1 is much simpler
than the one in table III.1, studied in the previous chapter. Most of the reaction rate coefficients in the
reaction scheme of table B.1 are constant, because they have been derived already considering Tg = 300
K. This original scheme does not take into account the recommendations of chapter III, since it preceded
it in time. However, we can notice that in table B.1 there is only the presence of charge transfer or ion
conversion in pure He from He+ towards He+
2 (R12), and not the other way around, which is consistent
with the conclusion derived in chapter III for atmospheric-pressure room-temperature He plasmas.
IV.2.1
Species evolution and kinetic schemes
Species evolution in the plasma
From zero-D simulations, we get the temporal evolution of the densities of the most chemically relevant
species, shown in figure IV.1. In this figure, we notice that the electron density after the passage of the
front has values above 1012 cm−3 and that quasi-neutrality is always attained between the electrons and
+
the ions He+ , He+
2 and N2 , which is mandatory due to the pre-ionization conditions and the reactions
considered. Both He ion densities, [He+ ] and [He+
2 ], arise during the ionization front and decrease quickly
to 108 cm−3 in about 100 ns and 130 ns, respectively. [He∗ ], composed of the densities of both excited
59
states He(23 S) and He(21 S), also arises during the electric field application but stays longer in postdischarge. N+
2 density, which increases both during E application and in post-discharge is clearly the
dominant ion starting from 40 ns into post-discharge.
Figure IV.1: Temporal evolution of the densities of the most chemically relevant species in the He plasma
with 1000 ppm of N2 using the reaction scheme of table B.1 and the zero-D model of section IV.1.
We can also use the zero-D simulations to better analyze the chemistry in the discharge. Figure
IV.2 shows the rates of creation of electrons, which are simultaneously the rates that provide the level
of ionization in the tube. One thing we can immediately notice from the reaction rates is that the
only ionizing reactions are due to direct electron impact or to the internal energy transfer from the He
metastables. During the electric field application, which corresponds to the passage of the ionization
3
∗
front, [He+ ], [N+
2 ], [N2 (C Πu )] and [He ] increase due to electron-impact collisions. Electron-impact
ionization is the most important ionization process in this region, particularly by He ionization. The E
field decreases from 10 ns until ∼ 20 ns but the growth of the densities by electron-impact is dominant
until ∼ 15 ns, when the field reaches ∼ 20 Td and species losses start compensating the creations by
+
electron-impact. He+
2 is created during the same time period, with about 2 ns of delay by respect to He ,
+
by ion conversion reaction He+ → He+
2 (R12). That is also the most responsible reaction for the He
loss. In post-discharge, the processes that have the biggest influence are no longer the electron-impact
collisions, since E is not high enough for electronic inelastic collisions to happen. Figure IV.2 also shows
the rates of production of the dominant ion in post-discharge N+
2 , which allows to better analyze what
happens during the post-discharge.
In the early post-discharge, between simulation times 20 ns and 100 ns, He+ and He+
2 are still very
important ions and ne is equal to the sum of the contributions of all 3 ions, no longer ∼ [He+ ] and not
yet ∼ [N+
2 ]. In this region, the species structure is changed mainly by ion conversion (R12) and by the
+
+
charge transfer reactions He+
2 → N2 (R13) and (R14). [He2 ] starts decreasing around 30 ns since there is
less He+ to create it by (R12) and charge transfer losses (R13) and (R14) take the upper hand. Although
He∗ is only created by electron-impact, its density seems to remain the same in this region, unlike [He+ ],
because He∗ losses are slower. We must notice that both the He ion conversion (R12) and the charge
60
Figure IV.2: Temporal evolution of the rates of production of e− (left), and of N+
2 (right), in the He
plasma with 1000 ppm of N2 using the reaction scheme of table B.1.
transfer reactions (R13-14) are not ionization reactions and therefore do not change the conductivity in
the channel but only the nature of the ions in the channel, even though they are the dominant reactions
in the early post-discharge.
Starting from 40 ns, N+
2 is the dominant ion and its density keeps increasing until ∼ 700 ns, thanks
to charge transfer in early post-discharge and, in the long term, to Penning ionization by He metastables
He∗ → N+
2 (R7) and (R8). It was calculated from the free software PumpKin (pathway reduction method
for plasma kinetic models) [72] that in the whole post-discharge period (t ≥ 20 ns) N+
2 is created with
contributions from Penning reactions (R7) and (R8) at 30% and 33%, respectively and from charge
transfer reactions (R13) and (R14) at 28% and 9%, respectively. By 100 ns, [He+ ] and [He+
2 ] have
+
significantly decreased and ne ' [N+
2 ]. Penning ionization not only helps N2 become the dominant ion
but, more importantly, it is clearly the main electron creation process in post-discharge, doubling the
ionization level from ∼ 1.5 to ∼ 3 × 1012 cm−3 in post-discharge. These reactions are also responsible for
∗
the loss of He∗ , since the creation of He+ and He+
2 from He by associative ionization (R5-6) are much
∗
less relevant reactions. The importance of Penning ionization of N+
2 sustained by He is corroborated
by references [29, 31–36]. Along with direct ionization (R1) and (R2), Penning ionizations are the most
important ionizing reactions that allow to have higher levels of ionization and conductivity in the channel
behind the streamer head. It is important to notice that in post-discharge, when N+
2 is the dominant
ion, the ionization is compensated by dissociative recombination of N+
2 (R11), which dominates from 1
µs forward.
In order to realize if the zero-D model describes the same reality for species evolution locally as the
2D model, figure IV.3 represents the distribution in space of the densities of the most relevant species at
the simulation time 1350 ns, obtained from the 2D simulations. From figure IV.3, we see that the ion
He+ only keeps high densities 1012 cm−3 on the front of the discharge. He+
2 also exists on the front of the
discharge but keeps densities of 1010 cm−3 for 2 cm behind the front. Metastable He∗ lasts longer and its
density decreases slowly between the discharge front and the anode. The ion that rests in the channel in
12
post-discharge is clearly N+
cm−3 . These results confirm that we can obtain a
2 with densities above 10
more detailed description of the species evolution in the 2D simulations by using the zero-D model.
61
Figure IV.3: Densities of the chemically relevant species in the He plasma with 1000 ppm of N2 at
simulation time 1350 ns using the reaction scheme of table B.1 and the 2D model of section I.3.1.
Re-evaluation of the reaction scheme
In order to better understand the influence of N2 in the dynamics of the discharge, we will focus on
understanding the evolution of species densities and reaction rates for different [N2 ]. Before doing so, we
first want to consolidate our confidence in the reaction scheme in use. Bibliographic references must be
used to chose reactions to study in our model. The reaction scheme from table B.1 is mainly based in
the conclusions from Pouvesle et al (1982) [29]. However, not all the reactions from [29] are included. A
list of further reactions that we wish to study can be found in table B.2 in appendix B.
Following the references [20, 21, 29, 37–40], we will no longer consider the He metastable state as the
sum of He(23 S) and He(21 S) but only as He(23 S), which reaches values 2 to 3 orders of magnitude greater
than He(21 S). He+
2 dissociative recombination existed already in table B.1 as (R10) with rate coefficient
9.0 × 10−9 cm3 s−1 . In fact, (R19) comes out as a subdivision of (R10) and the original coefficient of
(R10) is divided between the new (R10) (K = 4.0 × 10−9 cm3 s−1 ) and (R19) (K = 5.0 × 10−9 cm3 s−1 ).
Therefore, the introduction of (R19) does not consist in additional He+
2 loss but in a way of creating
He(23 S) without E field application, even though it provides a much lower rate than the electron-impact
excitation (R3) for every [N2 ] case tested. In fact, both (R10) and (R19) are much weaker in He+
2 loss
than charge transfer reactions (R13) and (R14). The association reaction (R20) is, along with (R4), one
of the only destruction processes of He(23 S), since the other loss processes consist of ionizations. It is
therefore a process that removes He(23 S) available for ionization. (R20) is always more relevant than
losses (R4), (R5) and (R6) but its importance by respect to Penning ionizations changes with [N2 ], being
62
far lower in the case studied so far, of 0.1% N2 . We can notice that the two reactions in table B.2 that
pay respect to He alone, which are (R19-20), were present in table III.1, although with different rate
coefficients.
2
Reference [29] introduces the excited ion N+
2 (B Σu ), created by electron-impact, Penning ionization
and He+
2 charge transfer. From these processes, only electron-impact, retrieved from the LXCat Phelps
database [52, 73] is actually a new ionization process, since Penning ionization and charge transfer are
obtained by dividing the coefficients from table B.1 between creation of ground state N+
2 (X) and excited
N+
2 (B). Dissociative recombination is also shared between the two species. Figure IV.4 shows the rates
+
18
of production of N+
cm−3
2 (B). Electron-impact ionization towards N2 (B) shows a weak rate (max 10
s−1 ) when compared to the other electron-impact ionization rates (R1) and (R2) (max 1021 cm−3 s−1 )
for every He-N2 mixture. However, references [23, 29, 71] point out that the radiative decay reaction
+
N+
2 (B) → N2 (X) + hν (R24) is, along with N2 (C) → N2 (B) + hν (R18), a factor of great importance
in experiments for emission spectroscopy diagnostics. In particular, N+
2 (B) is a monitor for the density
of He+
2 [29]. Figure IV.4 also illustrates the rates of light emission from these reactions (notice linear
y scale). We must notice that the rates have peaks at different simulation times, since N2 (C) comes to
exist during E application while N+
2 (B) has a longer creation process and a peak at a more distant time.
These time differences correspond to different positions in the tube behind the discharge front for a given
time. We conclude that it is important to include N+
2 (B) in the model for comparisons with experiments.
2
Figure IV.4: Temporal evolution of the rates of production of N+
2 (B Σu ) (left), and N2 (C → B) and
N+
(B
→
X)
light
emission
rates
(right),
in
the
99.9%
He
0.1%
N
plasma
using the reaction scheme
2
2
from tables B.1 and B.2.
+
Charge transfer reactions He+ → N+
loss, decreasing
2 (R21) and (R22) are very important in He
+
[He+ ] quicker than in the original scheme. N+
2 is produced earlier in time, when [He ] is high, due to
+
+
(R21) and (R22), but there is less He+
2 creation thanks to (R12), which means less He2 → N2 charge
transfer later in post-discharge. (R21) and (R22) do not change the e− production nor the [N+
2 ] in late
post-discharge but they change the moment when [N+
2 ] increases and the ion distribution in time. The
+
+
left side of figure IV.5 shows N+
2 creation rates in post-discharge that include He → N2 charge transfer
reactions and which can be compared with those of figure IV.2. Likewise, the result of changes so far
in the evolution of species densities can also be seen in figure IV.5 on the right-hand side and compared
63
with the results of figure IV.1.
Figure IV.5: Temporal evolution of the rates of production of N+
2 (left), and of the densities of the
chemically relevant species (right), in the 99.9% He - 0.1% N2 plasma using the reaction scheme from
tables B.1 and B.2.
By comparing these results with the ones before the introduction of the reactions in table B.2, we
can see that the variation of the density of the He metastable is not remarkable, which means (R19),
(R20) and the removal of He(21 S) had very low influence. He+ and He+
2 are lost slightly earlier due to
+
+
(R21) and (R22). [N+
2 ] is higher until 50 ns thanks to the earlier charge transfer He → N2 but is lower
+
between 50 ns and 500 ns, due to the share of He+
2 → N2 charge transfer and Penning ionization with
+
+
+
the new species N+
2 (B). Since N2 (B) is lost through de-excitation to N2 , by 500 ns we obtain ne = [N2 ],
as in the original case. We can also notice, specially in late post-discharge, that the electron density is
slightly higher than in the original case of figure IV.1. This is due to a slightly higher ionization during
the E field application through (R2b) but, more importantly, it is due to the fact that the recombination
coefficient (R11) of the dominant ion in post-discharge N+
2 was decreased, thanks to the introduction of
+
+
N+
2 (B). Addition of the N reactions from table 2, referenced from [29, 74] was also tested. N would
be created by electron-impact ionization (R25), by Penning ionization of N (R29) and from the share
+
17
of He+ → N+
cm−3 s−1 ) and
2 /N + N (R27-28). In fact, electron-impact ionization rates of N (∼ 10
Penning ionization rates of N (∼ 1014 cm−3 s−1 ) are very low and there is no relevant change in the
discharge structure by introducing N+ , whose density never attains the density level of the other ions of
1012 cm−3 .
Concluding about the evaluation of the reactions in table B.2:
• (R19) and (R20) are included in the model. Even though these reactions have low influence, the
reaction scheme is more consolidated with the inclusion of these creation and loss terms of He
metastables, by following the reference [29] and by coherence with the analysis of chapter III;
• He+ → N+
2 charge transfer reactions (R21-22) are also introduced, since they cause a significant
change in the ion structure in post-discharge;
• N+
2 (B) and the reactions that involve it are included, so that the model can furnish data about the
N+
2 (B) emission, in order to compare with experimental results;
64
• N+ and its reactions are not introduced. Since N+ is not leading to any relevant changes in the
species structure, we believe it is not worth it to include a new ion species with new transport
parameters in the 2D model.
∗
Besides the reactions in table B.2, the inclusion of an excited molecule He2 (a3 Σ+
u ), called He2 , and
the introduction of a stepwise ionization reaction He∗ + e− → He+ + 2 e− were considered. About
the excimer He∗2 , references [21, 31–35] seem to indicate that the effect of this metastable is to decrease
[He∗ ] and [He+
2 ] and then have the same ending as them, which is to provide energy for N2 ionization.
Stepwise ionization through He∗ was abandoned since reference [33] states that its influence is negligible
and other references [31, 32, 34, 35, 74] formulate these reactions in ways hardly compatible with the
current scheme. However, the study of chapter III has concluded that the stepwise ionization reactions
in table III.1 have a significant influence, which is why they should be added to the reduced scheme. We
have also considered the inclusion of more detail on N2 species and reactions, in order to be able to study
the plasma kinetics with higher N2 percentage in the He-N2 mixture. In this context, the same model and
applied electric field were used to study the kinetics in pure He using the simple reaction scheme in [22]
and in pure N2 using the simple reaction scheme in [75]. The consequence of that study was to add the
reactions in table B.3 to those of tables B.1 and B.2, in order to study the importance of including the
+
01 −
3 +
excimer He2 (a3 Σ+
u ), the metastables N2 (A Σu ), N2 (a Σu ) and the ion N4 and the reactions associated
to these species. Figure IV.6 depicts the densities of the relevant species in the plasma using the new
scheme, in the mixture we have been studying, 99.9% He - 0.1% N2 . Figure B.1 in appendix B represents,
for the same conditions, the rates of the electron-loss reactions and of the main Penning-type ionization
reactions through the internal energy of the excited species He∗ , He∗2 , N2 (A) (threshold energy 6.17 eV)
and N2 (a0) (threshold 8.52 eV).
Figure IV.6: Temporal evolution of the densities of the relevant charged species (left), and of some
chemically relevant species (right), using the reaction scheme from tables B.1, B.2 and B.3, in the 99.9%
He - 0.1% N2 plasma.
We note from the figures that He∗2 has a long life-time in the 99.9% He - 0.1% N2 mixtures and
therefore pushes Penning ionization of N2 longer in time through (R48) and moves the maximum of ne
to a higher instant in time, as stated in [74]. We should take into account that He∗2 is only created by
reactions (R20) and (R31). However, He∗2 does not bring much change to the plasma kinetics in the
65
mixture with 1000 ppm of N2 . For instance, while Penning ionization from He(23 S) reaches a rate of 1019
cm3 s−1 , the rate of reaction (R48) does not attain 1018 cm3 s−1 . In fact, the higher is [N2 ], the shorter
is the He∗2 presence and therefore, the smaller is its influence. It is, therefore, irrelevant in 97% He - 3%
N2 but relevant in He with 10 ppm of N2 for times higher than 10 µs, in which case it can change ne
in late post-discharge. As far as N+
4 is concerned, we notice in the figure that its density is irrelevant at
1000 ppm and that the recombination reaction associated to N+
4 (R33) has a much lower influence than
(R11). However, with 2500 ppm of N2 , N+
4 recombination is already important in late post-discharge and
in 97% He - 3% N2 , this ion is very relevant, becoming majoritary at 100 ns. But even in this latter case,
ionization from N2 (A) and N2 (a0 ) is irrelevant and even He∗2 ionization reactions (R47-49) are always
more important. In the particular case of interest of 1% N2 in the mixture, N+
4 recombination is higher
than (R11) and N+
4 is the dominant ion starting from 500 ns, being particularly important its creation
by N+
2 charge transfer (R32).
In conclusion from this analysis, it was decided to add N+
4 to the 2D model along with the associated
reactions, due to its importance in the He-N2 mixtures under study and in recombination in post-discharge
region, being already included in publications [27, 28]. However, the results from the zero-D model
∗
presented from here forward do not include N+
4 . On the contrary, the excimer He2 and the nitrogen
metastables N2 (A) and N2 (a0 ) were considered as less influent and were not included in the model, even
though the introduction of He∗2 and of the stepwise electron-impact ionization through He(23 S) may be
reconsidered in the future, taking into consideration the conclusions from chapter III.
Relevance of reactions on the discharge dynamics
The zero-D model also allows to study the direct influence of some reactions in the discharge dynamics.
The goal here is not to find out whether or not these reactions should be part of the model but to better
understand the phenomena in the discharge dynamics by isolating some reactions. The reaction scheme
of reference is the one explained earlier, based on the tables B.1 and B.2.
+
+
+
+
The influence of charge transfer reactions He+
2 → N2 /N2 (B) and He → N2 (R13), (R13b), (R14),
(R14b), (R21) and (R22) was studied by comparing the zero-D evolution with and without the inclusion
of these reactions. Figure B.2 in appendix B shows a direct comparison of the species temporal evolution
in the 2 cases for the 99.9% He - 0.1% N2 plasma. In fact, [He+ ] decreases slower but ends up by
+
disappearing by He+ → He+
2 ion conversion. On the other hand, if He2 is not subject to charge transfer
+
towards N+
2 , its density stays almost constant starting from 40 ns. In this case, N2 is only created in
post-discharge by Penning ionization and therefore never becomes the dominant ion. Instead, in post+
+
+
discharge we find ne = [He+
2 ] + [N2 ]. The difference is higher when He and He2 have more influence,
such as at low [N2 ], when these species last longer. Since these are not ionization reactions, not much
change in electron density is observed. Only in late post-discharge, by 10µs, it is possible to see that
ne is higher without charge transfer reactions because lower [N+
2 ] leads to less losses by (R11). In 2D
simulations, very small difference in the discharge dynamics was observed by removing the charge transfer
reactions.
+
The same study can be done for Penning ionization reactions He∗ → N+
2 /N2 (B) (R7), (R7b), (R8)
66
and (R8b). Figure IV.7 represents the species evolution without the inclusion of Penning reactions in the
model, which can be compared with the case in figure IV.5, where these reactions are included.
Figure IV.7: Temporal evolution of the densities of the chemically relevant species using the reaction
scheme from tables B.1 and B.2 without the Penning ionization reactions (R7), (R7b), (R8) and (R8b),
in the 99.9% He - 0.1% N2 plasma.
If the Penning reactions are removed, N+
2 is still the dominant ion in post-discharge thanks to charge
transfer reactions. However, there is no ionization in post-discharge and ne straightly decreases in this
region. Figure IV.8 depicts the effect that removing Penning reactions has on the 2D results of the
discharge propagation, by representing the spatial distribution of the electric field and of the electron
density at the same simulation time for the 2 cases, using the model presented in section I.3.1. In
fact, removing ionization in the post-discharge region significantly modifies the discharge dynamics. The
discharge propagates much slower and the channel has a much thiner structure. This result makes evidence
for the importance of Penning ionization reactions in post-discharge ionization, channel conductivity and,
therefore, discharge dynamics, as stated in [5]. It gives us an indication that for each He-N2 mixture with
low N2 addition, the discharge dynamics depends on the influence of Penning reactions.
The reaction scheme in use contains several reactions with the same effect with two bodies and with an
additional third He body. At atmospheric-pressure, with [He] near 2.45×1019 cm−3 , three-body reactions
are usually important [29]. The three-body reactions are the charge transfer and Penning reactions (R8),
(R8b), (R14), (R14b) and (R22). The same study of influence of the reactions was done for the three-body
reactions. In figure B.3 in appendix B, the zero-D results for the temporal evolution of the most relevant
species are represented including and removing the three-body reactions from the reaction scheme, in
the 99.9% He - 0.1% N2 plasma case. The influence of these reactions is only quantitative. Less charge
transfer and Penning ionization reactions will happen but the discharge structure remains the same. As
far as the ionization by Penning reactions is concerned, [He(23 S)] decreases slower if there is no three-body
reaction. Instead of [He(23 S)] decreasing to 108 cm−3 around 2.5µs, it happens around 5µs. Therefore,
[e− ] and [N+
2 ] rise later. Instead of a peak around 700 ns, these quantities would have their maximum
around 1µs if there were no three-body reactions.
67
Figure IV.8: 2D simulation results for the propagation of the 99.9% He - 0.1% N2 plasma at 2 µs
simulation time, using the reaction scheme from tables B.1 and B.2 with Penning reactions (left), and
without Penning reactions (right), from the model presented in section I.3.1.
IV.2.2
Influence of N2 concentration in the He-N2 plasma
Influence of N2 concentration in the kinetics of a He-N2 plasma
Using the new reaction scheme (tables B.1 and B.2), simulations were carried out for different N2 concentrations between 10 ppm and 30 000 ppm (97% He - 3% N2 ). The case with the lowest N2 concentration
considered is 10 ppm, since it is not credible that there would exist pure He at room temperature and
at atmospheric pressure and, as we remark in figures II.9, A.10 and II.11, the EEDFs and the electronimpact rate coefficients in pure He are already very close to those in He with 100 ppm of N2 , which is
why we believe nothing would be gained from studying cases with lower [N2 ] than 10 ppm.
In order to understand the influence of N2 concentration, on the left side of figure IV.9 we observe how
species evolve in case there are 10 ppm of N2 in the mixture. We can compare these results with those of
figure IV.5 for 1000 ppm of N2 . In fact, there is no big difference in the electron-impact He excitation and
ionization during the electric field application. However, there is a difference of two orders of magnitude
of the electron-impact N+
2 creation rate that influences the level of ionization from the moment the E
field is applied to the plasma. This result is consistent with the fact that at low fields the direct ionization
coefficient is slightly higher for 1000 ppm of N2 than for lower concentrations, as seen in figure II.11.
However, it is mainly due to the lower level of N2 density participating in the N2 ionization reactions.
With less [N2 ], the losses of He+ and He+
2 by (R13), (R14) and (R21), (R22) are much lower and these
+
species last longer in time. However, charge transfer from He+
2 towards N2 will still act slowly, being the
+
most responsible reactions for the N+
2 increase starting at t = 20 ns. He2 becomes the dominant ion in
early post-discharge as it is only surpassed by N+
2 around t = 2 µs. Most importantly, due to the low [N2 ],
the rates of Penning reactions (R7) and (R8) are much lower and ne never increases in post-discharge as
it does in the case with 1000 ppm of N2 . There is a remarkable difference between the two cases in the
ionization level between 10 ns and 10 µs.
We can also study the influence of N2 concentration in a case with [N2 ] higher than 1000 ppm. Figure
68
Figure IV.9: Temporal evolution of the densities of the most chemically relevant species for the cases of
10 (left) and 10 000 (right) parts per million of N2 in the He-N2 plasma, using the reaction scheme from
tables B.1 and B.2.
IV.9 also represents on its right-hand side how species densities evolve in the case of 1% of N2 . At the
order of the unity of percentages of the [He]/[N2 ] variation, the change in the He density starts to also
have influence and, in fact, during the field application in the 99% He - 1% N2 plasma, there is an increase
of approximately 10 times in N2 electron-impact ionization rates but there is a higher decrease in He
ionization and in He excitation rates, with respect to the reference case of 99.9% He - 0.1% N2 of figure
IV.5. This is consistent with the results in figure II.11, where it is shown that global electron-impact
ionization is lower in the 1% N2 case than for lower concentrations of N2 . Lower [He+ ] will generate
+
less [He+
2 ] by (R12) and charge transfer reactions will have very little significance, as N2 becomes the
dominant ion still during the E field application. Thanks to the N2 density, Penning ionization is very
relevant in the 99% He - 1% N2 plasma in early post-discharge starting from 20 ns instead of 80 ns like
in the 0.1% case. ne actually increases in early post-discharge with 1% N2 , attaining a maximum around
100 ns instead of 1 µs in the 0.1% case and 12 ns in the 10 ppm case. However, as He excitation is lower
during the discharge, He(23 S) is lost quicker with 1% N2 than with lower [N2 ] and, therefore, ne does not
double in post-discharge as it did in the 0.1% case. This result shows the relevance of the He excitation
coefficient, with a similar shape with the ionization coefficient of figure II.11 and which is significantly
lower in 99% He - 1% N2 plasmas than in 99.9% He - 0.1% N2 plasmas at low E/N fields. The evolution
of species densities for the cases of 100 and 30 000 ppm of N2 in the mixture are depicted in appendix B,
figure B.4. We must notice that for higher values of [N2 ] the reaction model used may no longer be valid,
since it was obtained from bibliographic references considering N2 as impurities in small concentrations.
In figure IV.10 we take a look at the general picture, by observing the spatial distribution of the
densities of N2 (C), using the model presented in section I.3.1, which are representative of what light
emission would be in experiments, represented for the same time of simulation and the same conditions
(applied voltage to the anode Va = 6 kV). The figure allows us to notice the position of the N2 (C)
densities, which is the position of the ionization front. The conclusion to take from this figure is that the
discharge propagation is faster and larger for N2 concentration in the order of magnitude of 1000 ppm.
In fact, there is a big increase of dynamics between the small concentrations of N2 of 10 or 100 ppm and
69
1000 ppm and there is a small decrease between 0.1% and 1% of N2 in the mixture. These results agree
qualitatively with those obtained experimentally in [23], presented in figure I.4 where there is a change in
velocity with a maximum for 2500 ppm of N2 and a change of the discharge structure with the variation
of the N2 concentration.
Figure IV.10: Two-dimensional spatial distribution of the N2 (C 3 Πu ) densities at t = 2000 ns for 10, 100,
1000 and 10 000 parts per million of N2 in the He-N2 plasma, using the reaction scheme from tables B.1
and B.2 and the model presented in section I.3.1.
We want to understand why the discharge dynamics changes with [N2 ], as seen in figure IV.10. In
fact, if we compare the ionization levels (electron densities), depicted in figure IV.11, we see that the
ionization is in fact higher in post-discharge for the 1000 ppm of N2 case. Moreover, for high [N2 ] in the
gas mixture, such as the 97% He - 3% N2 case, there is a huge difference in the ionization level. These
results can be compared with those from the 2D simulation along axial positions in figure B.5 in appendix
B. However, the analysis by the e− densities is not enough to explain why figure IV.10 presents such a
big difference in speed between the 10 ppm and the 1000 ppm of N2 cases and why that difference is so
small between the 0.1% and the 1% N2 cases.
Looking at the most influent electron source reaction, Penning ionization, represented in figure IV.11
for the different [N2 ], we can see that electron creation in the early post-discharge is significantly higher
for 1000 or 10 000 ppm of N2 than for lower concentrations. Our conclusion from this study is that the
dynamics of propagation not only depends on the ionization by the electric field front but also depends
greatly on Penning ionization in early post-discharge, before 1 µs after the field passage. Extrapolating to
the plasma in the tube, ionization in the channel behind the discharge front is of great importance to the
70
Figure IV.11: Comparisons for several values of [N2 ] in the He-N2 plasma of the temporal evolution of
the electron/ion densities (left), and of the creation rates by Penning ionization (R7) (right), using the
reaction scheme from tables B.1 and B.2.
discharge propagation. In fact, the ionization behind the front increases the conductivity of the channel,
which approaches the potential in the anode to the one in the discharge head and therefore increases the
potential difference between the head and the end of the tube, which creates the local electric field that
leads the discharge propagation.
We must notice that in the 2D simulations of the discharge in the tube, the electric field in the
front does not necessarily have the same maximum and shape in all the cases. The field depends on the
potential drop in the channel and therefore varies with time and with the mixture. This means that the
zero-D study gives approximate results of densities and rates, since the same E field (estimated for the
1000 ppm case) was applied for all mixtures. However, for this case, as seen in figure B.6 in appendix B,
the longitudinal electric field in the head has got almost the same maximum and shape for the different
mixtures at the same simulation time, which means that the changes verified with [N2 ] in the 2D results
from figure IV.10 are due to the local chemistry we have been studying. In fact, it seems that if the
conductivity in the channel is higher, the field has higher values in a determined position, which leads the
front to propagate faster and, therefore, have a more advanced position at a given instant in time. We
are lead to conclude that the conductivity in the channel that connects the potential in the anode at the
start of the tube to the potential in the discharge front is of great importance to the front propagation.
The zero-D results allow to explain the desired phenomena, as long as we know how to isolate them.
Influence of N2 concentration on the pre-ionization of He-N2 plasmas
In this chapter, plasmas were not considered as first discharges in a tube but as part of a repetition of
discharges, which is why there is a pre-ionization of electrons and ions N+
2 . For all studied conditions
in this work, we have considered a level of pre-ionization of 109 cm−3 . It is now required to discuss the
value of the pre-ionization level. In fact, it has been seen previously, as in figure IV.11, that although in
late post-discharge ne = [N+
2 ], the ionization value is not the same for every case at the end of the scale
considered, 10 µs. However, as time is extended further, we notice in figure IV.12 that the ionization
levels reach the same values for every case (3 very distinct cases of 10, 1000 and 30 000 ppm are shown)
71
for times higher than 100 µs.
Figure IV.12: Temporal evolution of the electron density, with pre-ionization 109 cm−3 of e− and N+
2,
using the reaction scheme from tables B.1 and B.2, for different [N2 ] in the He-N2 discharge: 10, 1000
and 30 000 ppm.
In late post-discharge, when the only charged species are electrons and N+
2 , the dominant reaction
−
that leads to charge loss is the dissociative recombination reaction N+
→ 2 N (R11), for all the
2 + e
He-N2 mixtures considered in this chapter. At that point, since ne = [N+
2 ], the electron loss equation is
formulated as:
∂ne
= −Krecombination n2e
∂t
1
1
1
ne (t) =
'
f or t ≥
Krec t
Krec n0e
Krec t + n10
(IV.2)
(IV.3)
e
Where n0e is the electron density at the moment when we take (R11) as the dominant reaction.
As (R11) dominates and time increases, the evolution of electron densities gets to depend only on the
rate coefficient, which we take as constant. Therefore, the electron density stops depending on an earlier
ne value and thus on the N2 concentration. We must notice that this zero-D model may not contain all the
particle balance processes that have influence in long time-scales, in late post-discharge, such as diffusion
losses. However, the 2D simulations show that the charges survive at least 2 µs in post-discharge (no
simulation results for higher times) and we do believe that the dissociative recombination is the dominant
process at atmospheric-pressure, which makes the kinetics in late post-discharge equal in all mixtures.
The pre-ionization is independent of the N2 concentration in the mixture as long as the time between
pulses exceeds some 100s µs, which in case of repetitive pulses means, for applied frequencies lower than 1
kHz. The experiments we have been making comparisons with, from the GREMI group [23], use a pulser
working at 500 Hz, which is within the applicable range. Unlike what is stated in [30] for a different
applied electric field, here Penning ionization is not responsible for the pre-ionization in the plasma but
dissociative recombination alone is.
However, results show that the pre-ionization is not independent of the frequency of the pulsed applied
voltage. In order to find at which time the e− density reaches the pre-ionization level, which is called
72
here tM AX , equation (IV.3) is solved, providing the result in eq. (IV.4). tM AX is the time when a
repetitive pulse should be applied to preserve the pre-ionization level. In fact, in figure IV.13 we can see
that tM AX follows that result and that in the same mixture, for different pre-ionization levels chosen,
there is a different tM AX proper for pulse application. Therefore, a frequency can be defined so that a
pre-ionization is fixed along a whole cycle of pulses.
ne (tM AX ) = ne0 <=> tM AX =
1
Krec ne0
f = Krec ne0 , with Krec = 1011 cm3 s−1
(IV.4)
(IV.5)
Figure IV.13: Temporal evolution of the electron density, using the reaction scheme from tables B.1
and B.2, in the 99.9% He - 0.1% N2 plasma with different pre-ionization values of e− and N+
2: 5 ×
109 , 1010 , 1011 , 1012 cm−3 .
As we know, not all the loss processes are included in the zero-D model and therefore results are not
quantitatively rigorous. Also, we are assuming the same electric field application in the discharge front
for all the cases of pre-ionization, which may not be true due to the conductivity of the channel. But this
estimation points out that if one intends to use a 100 kHz pulser, a pre-ionization around 1012 cm−3 of
10
electrons and of N+
cm−3 of ne0 should
2 should be assumed and if one intends to use a 1 kHz pulser, a 10
be assumed. If a pulser with higher frequency than 100 kHz is used, which implies a time between pulses
below 1 µs, these estimations are no longer possible, since the late post-discharge is not attained and the
relation in eq. (IV.3) is no longer valid. For the current case, we believe that the assumption of a 109
cm−3 pre-ionization is a good estimation if we intend to compare results with experimental conditions
with pulse frequencies between some Hz and 1 kHz.
73
74
Chapter
V
Conclusions and future work
During the development of the master thesis, I have worked on the kinetics of Helium-based plasmas
at atmospheric-pressure, focusing on the ionization mechanisms in those plasmas. In particular, I have
studied electron kinetics in Helium plasmas with Nitrogen, Oxygen and dry-air (80% N2 - 20% O2 )
impurities, heavy species kinetics in pure Helium discharges at steady-state and the kinetics of He plasmas
with N2 impurities propagating at atmospheric pressure in dielectric tubes.
I have used and developed the IST-LoKI (LisbOn KInetics) code of the GEDG (Gas Discharges and
Gaseous Electronics) group of IPFN (Instituto de Plasmas e Fusao Nuclear), in order to include Helium
to the list of gas in which IST-LoKI can perform kinetics calculations. The Electron Boltzmann Equation
(EBE) solver of IST-LoKI is now able to perform calculations in Helium, as well as in Argon, Nitrogen,
Oxygen and Hydrogen. This solver has allowed to obtain the electron energy distribution functions
(EEDF) and all the plasma parameters calculated from the EEDF in Helium. Good results of power
balance and correct fits of experimental data were obtained for transport parameters and for electronimpact rate coefficients in Helium for reduced electric fields E/N ≤ 100 Td. These results were also
compared with the calculations of other numerical tools, LXCat and Bolsig+. The limitations of the
IST-LoKI code without secondary electron production and energy sharing for high E/N were noticed.
Furthermore, for the first time, the densities of excited states were introduced in IST-LoKI as separate
species from the ground-state species. This approach seems correct, as good results of power balance were
obtained for these cases. The effects of low concentrations of the excited states He(23 S) and He(21 S) on
the EEDF were analyzed. On the one hand, the introduction of these states adds new excitation and
ionization channels for the electrons in the plasma. On the other hand, these states carry the introduction
of superelastic collisions that provoke a plateau in the EEDF and its depletion towards higher kinetic
energies. These effects have consequences in the plasma parameters and particularly on the ionization
coefficient, where the effect of stepwise ionization is remarkable at fields E/N ≤ 20 Td.
The electron distribution functions and the swarm parameters were also studied for several binary
mixtures of He-N2 and He-O2 and ternary mixtures He-air, where we take synthesized air as 80%N2 20%O2 . Focus was given to the cases of small concentrations of the gases N2 , O2 and air, between 10
parts per million and 10%. The EEDFs have been compared for the different mixtures, exhibiting higher
75
tails in the pure noble gas Helium. The changes in the EEDF have a noticeable effect in the calculated
electron-impact rate coefficients. Among these, emphasis was given to the global ionization coefficient
and its variation with the mixture and with the electric field. The cases with low molecular gas densities,
around 0.1% (1000 parts per million), were shown to provide higher ionization coefficients than the case
of pure He, due to the low ionization thresholds of N2 and O2 . The electronegativity of O2 has proved
to affect the electron kinetics, as a result of the electron attachment mechanism. The analysis of the
effective ionization rate coefficient (Kion − Katt ) shows that discharge breakdown is more difficult in the
presence of O2 .
The solver of the particle rate-balance equations of IST-LoKI has also been developed to include a
reaction scheme for pure Helium. The scheme used was a reduced version of a larger scheme of reference.
The results from this model were compared with those using the complete scheme and a different numerical
tool, for similar plasma conditions. Significant differences were noticed, particularly in the densities of
the ions He+ and He+
2 . This analysis has allowed to notice the limitations of the reduced scheme and
to point out the importance of including the atomic excited states He(n > 2, l, s) and the associative
−
ionization reactions He(n > 2, l, s) + He → He+
2 + e .
Then, the collisional-radiative model (CRM) using the reduced scheme and the tool IST-LoKI has
been used to simulate the plasma kinetics in a steady-state discharge with fields E/N between 4 Td and 7
Td and electronic densities ne between 109 and 1014 cm−3 , at atmospheric-pressure, at room-temperature
Tg = 300 K and in a tube with an inner radius R = 2 mm. The differences between the results obtained
for different electron densities and electric fields have been noted and conclusions have been taken about
which species and reactions are negligible and can be taken out of the model, in case a more reduced
scheme is intended. In fact, the excited species He(21 P ) and its associated reactions, along with the
+
electron-assisted recombination reactions and the He+
2 → He ion conversion were considered irrelevant
for the kinetics of charged species in this plasma. However, the results have also allowed to notice that
the dominant ion species in this pure Helium plasma is the molecular He+
2 and that direct, stepwise,
Penning and excimer ionization reactions are essential to be included in a Helium Patm model to describe
charge production, just as dissociative recombination and diffusion processes are required to describe
charge loss.
Taking these results into account, the next step in the development of the Helium scheme in ISTLoKI will be the implementation of a more complete scheme with the inclusion of He(n > 2, l, s) and
−
the associative ionization reactions He(n > 2, l, s) + He → He+
2 + e . Then, the study of the model for
different electronic densities can be repeated and the means to include a single state He(n > 2) can be
studied. Even though the conclusions taken from the reduced scheme will likely remain valid, a new study
with the complete scheme can provide new information on the important species and kinetic processes
in the plasma.
During my internship in the EM2C laboratory (Énergétique Moléculaire et Macroscopique, Combustion) I have worked on the propagation of plasmas at atmospheric pressure in dielectric tubes and I have
focused on the kinetics of He plasmas with N2 impurities. This work was also included in this master
thesis.
76
I have determined transport parameters and electron-impact rate coefficients for several He-N2 mixtures with N2 concentrations from 10 to 100,000 ppm and I have written the subroutine to implement
these coefficients in a 2D discharge code. The results show that the variation of electron-impact rate
coefficients with [N2 ] in the He-N2 mixture is very important for the discharge dynamics, in particular
thanks to the variation of the global electron-impact ionization and the He electron-impact excitation
towards the He metastable state He(23 S), a species that has an important role in ionization, particularly
in post-discharge.
Then, I have implemented a zero-dimensional plasma kinetics model using ZDPlasKin. Particle rate
equations have been solved for a reduced He-N2 kinetic scheme, considering the local application of a
transitory electric field with 50Td maximum amplitude. These conditions are very different from those
explained earlier using IST-LoKI. The reactions from the original reaction scheme of the 2D model have
been studied, bibliographic references have been followed and other reactions have been tested. From
+
+
this study, charge transfer He+ → N+
2 reactions and the ions N2 (B) and N4 were found important to
add to the reaction scheme. Still, for the future, the inclusion of the metastable molecule He∗2 and of the
stepwise ionization reaction using He(23 S) may be considered.
Coupling zero-D and 2D model results allowed us to understand the role of species evolution and reactions in the discharge dynamics. The relation between nitrogen admixture and the different elementary
processes participating in the kinetics has been studied and discussed, revealing the influence of N2 on
the plasma ionization level and its implications in the discharge structure. It was found that discharge
dynamics and structure changes a lot with N2 concentration in the mixture. In particular, there is a
much faster and larger propagation in the tube if the mixture contains 1000 ppm of N2 rather than 100
ppm. The two-dimensional simulation results for discharge dynamics and structure provide qualitative
agreement with experiments. Very importantly, by the zero-D results and by their coupling with the 2D
results, the key role of kinetic processes, especially Penning reactions, on the ionization degree of the
plasma has been put forward. It was found that not only the ionization created during the discharge by
electric field application but also the ionization in post-discharge by Penning reactions provide explanation to the difference in dynamics with the variation of [N2 ] in the He jet propagating along the tube.
As the influence of Penning reactions He(23 S) → N+
2 was made evident, we conclude that ionization in
post-discharge in the channel is extremely important for the discharge dynamics. Consequently, we are
lead to conclude that the conductivity in the channel that connects the potential in the anode at the
start of the tube to the potential in the discharge front is of great importance to the front propagation.
Finally, I have studied how the modeled pre-ionization in the tube should be defined by respect to the
[N2 ] variation to find that there is no variation of pre-ionization with [N2 ] as long as we are considering
results for repeated pulses with frequencies below 1 kHz. This happens thanks to the supremacy of the
−
dissociative recombination reaction N+
→ N + N in late post-discharge that makes kinetics in
2 + e
late post-discharge equal to all mixture cases. In fact, the pre-ionization dependence on frequency was
estimated to be ne0 = f /Krec , as long as we are considering frequencies below 100 kHz. The pre-ionization
in use, ne0 = 109 cm−3 was found to be a good estimation in case we are considering frequencies between
some Hz and 1 kHz and He-N2 mixtures between 10 ppm of N2 and 95% He - 5% N2 .
77
Combining the experience of using both numerical tools IST-LoKI and ZDPlasKin, I have concluded
that ZDPlasKin has a more practical application but, as a developer, I have more freedom and more
understanding of the calculations and of the implied physics using IST-LoKI, which awards it a higher
potential. Therefore, I propose the sequence of this master thesis to be connected to the development
of IST-LoKI. This code should be modified to allow an easier and more universal input of data and to
become more user-friendly. Furthermore, it can be used to continue the study of kinetics in He-based
plasmas. Firstly, the code can be easily modified to include the study of post-discharge. This means
that after obtaining the results for the steady-state discharge, the electric field can be set to zero and
the temporal evolution of the species populations can be studied. Then, if the temporal duration of the
discharge is shortened, for instance, to the µs range, we can change from a regime of steady-state to a
pulsed discharge and approach the conditions of the plasma jets propagating in dielectric tubes. Moreover,
the code can still be altered to include the possibility of an input temporal electric field and free electron
density, in which case the IST-LoKI can be used like ZDPlasKin with a transitory electric field. These
changes will allow the study of the collisional-radiative model for pure Helium in more diverse conditions
and the consolidation of the recommendations about a reduced kinetic scheme. Then, the code can be
used to study the plasma kinetics in He-N2 , He-O2 and He-air mixtures. The reaction schemes already
present in IST-LoKI for He, N2 , O2 and N2 -O2 can be used and bibliographic references (presented in
this thesis) can provide hypothesis for complete reaction schemes for He-N2 and He-O2 mixtures. Once
again, reduced schemes can be deduced from these studies for the diverse cases and plasma conditions
and results can then be compared with results from ZDPlasKin itself and coupled with results of 2D
simulations and of experiments.
At a personal level, I should state that I have applied knowledge from my universitary formation
from both the IST in Lisboa and the M2 PEL in Paris during the development of the master thesis. I
have participated in teamwork dynamics and I have worked with different people, laboratories and work
conditions. I have acquired competences and knowledge on the subjects of plasma jet propagation and
its coupling with medical applications and on the kinetics of electrons and heavy species in Helium-based
plasmas at atmospheric-pressure. I have had the opportunity to work with different numerical tools,
among which the plasma community software LXCat, Bolsig+, ZDPlasKin and IST-LoKI. Finally, I
have acquired experience in zero-D plasma modeling, in the coupling of zero-D/2D results and in the
comparison of simulation and experimental results.
78
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84
Appendix
A
Electron kinetics in Helium-based plasmas
In figure A.1, the cross-sections of elastic collisions between electrons and ground-state He, of excitation
collision e− + He(11 S) → e− + He(23 S) and of ionization collision e− + He(11 S) → He+ + 2 e− are
shown. Linear interpolation was always used.
Figure A.1: Input cross-sections and interpolated cross-sections used for calculations for elastic collisions,
excitation collisions and ionization collisions between e− and He(11 S) using linear interpolation.
In figures A.2 and A.3, the same plots of figure II.1 are represented with more detail, comparing
the calculated and the Maxwellian distributions for each E/N . The power balance resulting from the
85
calculation of these EEDFs is described in section II.2.1.
Figure A.2: Results for the calculated and Maxwellian EEDFs when the gas is completely ground-state
Helium in logarithmic scale for different input reduced electric fields 1 Td (uM AX =60 eV, N=6000), 10
Td (uM AX =125 eV, N=7500), 50 Td (uM AX =250 eV, N=7500) and 100 Td (uM AX =550 eV, N=7500).
Figure A.3: Results for the calculated and Maxwellian EEDFs when the gas is completely ground-state
Helium in logarithmic scale for different input reduced electric fields 250 Td (uM AX = 1000 eV, N =
7500), 500 Td (uM AX = 1000 eV, N = 7500) and 1000 Td (uM AX = 1000 eV, N = 7500).
86
Figure A.4 presents the results of swarm parameters µe × N , De × N and uK for N2 and O2 from
LoKI, from the several models of Bolsig+ and from experimental results in LXCat [52].
Figure A.4: Swarm parameters for pure N2 and pure O2 as function of E/N from several models and experiments: reduced electronic mobility, reduced electronic diffusion coefficient and electron characteristic
energy.
Figure A.4 shows that the results from LoKI generally agree with those from Bolsig+ and that both
show the same trends as the experimental values. But it is also shown that there are always higher
deviations from the experimental values considering the diffusion coefficient and that, in general, the
LoKI results follow better the experimental values at low E/N .
Figure A.5 presents the results of swarm parameters µe × N and uK for dry-air (80%N2 -20%O2 ) from
87
LoKI and for real air from experimental results from [53]. Experimental results of the characteristic
energy were obtained after removing CO2 from the real air.
Figure A.5: Reduced electronic mobility (left) and electron characteristic energy (right) as function of
E/N for air from experiments and for dry-air from LoKI.
The results from figure A.5 show a good agreement between the calculations and the experimental
values, although there are few experimental values for some ranges of E/N .
Figure A.6 shows the most relevant cross-sections affecting the EEDF shape.
Figure A.6: Most relevant collision cross-sections between electrons and He states.
Figure A.7 shows the effect of adding metastable densities [He(21 S)] and [He(23 S)]=[He(21 S)] on the
88
EEDF for E/N =1 Td.
Figure A.7: Results for the calculated EEDFs in logarithmic scale for E/N =1 Td and several mixtures
of He(11 S) and He(21 S) (left) and He(11 S), He(23 S) and He(21 S) (right).
Figure A.8 presents the swarm parameters De × N , µe × N and uK for varying ground-state He and
He(23 S) densities, calculated for a range of E/N between 0.01 Td and 300 Td.
Figure A.8: Swarm parameters for different mixtures of He(11 S) and He(23 S) as function of E/N : reduced
electronic diffusion coefficient De × N , reduced electronic mobility µe × N and electron characteristic
energy uK .
Results of swarm parameters De × N , µe × N , uK , α/N and total ionization coefficient Kion obtained
89
for several E/N until 300 Td and for the cases of pure ground-state He and He with relative densities
of 10−4 of He(23 S), He(21 S) and both are presented in figure A.9. It is not usual to calculate or
measure swarm parameters for excited-state gases and, therefore, both figure A.9 and figure A.8 present
no comparisons with other results.
Figure A.9: Swarm parameters for different Helium states mixtures as function of E/N : reduced electronic
diffusion coefficient De × N , reduced electronic mobility µe × N , reduced Townsend ionization coefficient
α/N , ionization coefficient Kion and electron characteristic energy uK .
As far as the reduced electronic diffusion coefficient, the reduced electronic mobility and the characteristic energy are concerned, we can notice from figure A.9 that the effects of adding one excited state
or the other are very similar. At intermediate fields (5 Td < E/N < 20 Td), the presence of both excited
90
states seems to have an effect of addition of the contribution of each excited state. However, we notice
that for lower fields the results are only different from the ground-state results if there is the presence of
both excited states.
Figures A.10, A.11 and A.12 compare EEDFs for different He-N2 , He-O2 and He-air mixtures, respectively, for fields E/N = 1 Td (extreme case below the reference fields ∼ 5 Td in [12]) and E/N = 50 Td
(same order of magnitude as the maximum field ∼ 40 Td of section I.3.1).
Figure A.10: Results for the calculated EEDFs in logarithmic scales for E/N = 1 Td (left) and E/N =
50 Td (right) and several mixtures of He(11 S) and N2 .
Figure A.11: Results for the calculated EEDFs in logarithmic scales for E/N = 1 Td (left) and E/N =
50 Td (right) and several mixtures of He(11 S) and O2 .
91
Figure A.12: Results for the calculated EEDFs in logarithmic scales for E/N = 1 Td (left) and E/N =
50 Td (right) and several mixtures of He(11 S) and dry air.
Figure A.13 shows the results of calculated EEDFs at 10 Td and 50 Td for He-O2 mixtures, considering
excited He with relative densities 10−4 of He(23 S) and He(21 S).
Figure A.13: Results for the calculated EEDFs in logarithmic scales for E/N = 10 Td (left) and E/N
= 50 Td (right) and several mixtures of He(11 S), He(23 S), He(21 S) and O2 , with He(23 S) and He(21 S)
relative densities 10−4 .
92
Appendix
B
Zero-dimensional simulation of an
atmospheric-pressure He-N2 tube streamer
Nr
R1
R2
R3
R4
R5
R6
R7
R8
R9
R10
R11
R12
R13
R14
R15
R16
R17
R18
Table B.1: Original kinetic scheme for a He plasma with 1000 ppm of N2 from [26].
Process
Reaction
Rate coefficient
Direct ionization
He + e− → He+ + 2 e−
f (E/N )
−
N2 + e − → N+
+
2
e
f (E/N )
2
Direct excitation
He + e− → He∗ (23 S/21 S) + e−
f (E/N )
De-excitation
He∗ + e− → He + e−
2.9 × 10−9 cm3 s−1
Associative ionization
He∗ + He∗ → He+ + He + e−
4.35 × 10−10 cm3 s−1
+
∗
∗
−
He + He → He2 + e
1.015 × 10−9 cm3 s−1
+
∗
−
Penning ionization
He + N2 → He + N2 + e
7.6 × 10−11 cm3 s−1
+
∗
−
3-body Penning
He + N2 + He → 2 He + N2 + e
3.3 × 10−30 cm6 s−1
T
+
−
−
−
Recombination
He + e + e → He + e
8.0 × 10−20 ( Tge )4 cm6 s−1
+
−
−9
Dissociative recomb.
He2 + e → He + He
9.0 × 10 cm3 s−1
+
−
N2 + e → N + N
9.0 × 10−9 cm3 s−1
+
+
Ion conversion
He + 2 He → He2 + He
1.5 × 10−31 cm6 s−1
+
+
Charge transfer
He2 + N2 → N2 +2 He
1.1 × 10−9 cm3 s−1
+
+
He2 + N2 + He → N2 +3 He
1.36 × 10−29 cm6 s−1
−
3
−
Direct excitation
N2 + e → N2 (C Πu ) + e
f (E/N)
De-excitation
N2 (C 3 Πu ) + N2 → N2 + N2
8.0 × 10−11 cm3 s−1
N2 (C 3 Πu ) + He → N2 + He
1.0 × 10−12 cm3 s−1
3
3
Radiative decay
N2 (C Πu ) → N2 (B Πg ) + hν
2.4 ×107 s−1
93
Reference
[17, 66]
[17, 66]
[17, 66]
[29]
[29]
[29]
[29]
[29]
[30]
[29]
[29]
[29]
[29]
[29]
[17, 66]
[76]
[76]
[76]
N br
R19
R20
R21
R22
R23
R2b
R7b
R8b
R11b
R13b
R14b
R24
R25
R26
R27
R28
R29
R30
N br
R31
R32
R33
R34
R35
R36
R37
R38
R39
R40
R41
R42
R43
R44
R45
R46
R47
R48
R49
R50
Table B.2: List of proposed reactions to add to those of table B.1.
Process
Reaction
Rate coefficient
+
−
3
Dissociative recomb.
He2 + e → He(2 S) + He
5.0 × 10−9 cm3 s−1
3
∗
Association
He(2 S) + 2 He → He2 + He
1.9 × 10−34 cm6 s−1
+
+
Charge transfer
He + N2 → He + N2
1.2 × 10−9 cm3 s−1
+
+
He + N2 + He → 2 He + N2
2.2 × 10−29 cm6 s−1
N Association
N + N + He → N2 + He
1.15 × 10−29 cm6 s−1
+
−
−
Direct ionization
N2 + e → N2 (B) + 2 e
f (E/N )
−
Penning ionization
He(23 S) + N2 → He + N+
0.5 × 7.6 × 10−11 cm3 s−1
2 (B) + e
−
3-body Penning
He∗ + N2 + He → 2 He + N+
0.5 × 3.3 × 10−30 cm6 s−1
2 (B) + e
+
−
Dissociative recomb.
N2 (B) + e → N + N
0.5 × 2 × 10−7 cm3 s−1
+
+
Charge transfer
He2 + N2 → N2 (B) +2 He
0.25 × 1.1 × 10−9 cm3 s−1
+
+
He2 + N2 + He → N2 (B) + 3 He
0.25 × 1.36 × 10−29 cm6 s−1
+
+
Radiative decay
N2 (B) → N2 (X) + hν
1.5 × 107 s−1
−
+
−
Direct ionization
N+e →N +2e
f (E/N )
Dissoc. ionization
N2 + e − → N+ + N + 2 e −
f (E/N )
Charge transfer
He+ + N2 → N+ + N + He
0.5 × 1.2 × 10−9 cm3 s−1
He+ + N2 + He → N+ + N + 2 He
0.5 × 2.2 × 10−29 cm6 s−1
3
+
−
Penning ionization
He(2 S) + N → N + e + He
1.6 × 10−10 cm3 s−1
+
−
Recombination
N +e →N
5.0 × 10−9 cm3 s−1
Table B.3: List of proposed reactions to add to those of tables B.1 and B.2.
Reaction
Rate coefficient
Reference
+
∗
−9
3 −1
He+
+
N
→
N
(B)
+
He
0.75
×
1.1
×
10
cm
s
[29, 33]
2
2
2
2
+
−29
6 −1
N+
+
2
N
→
N
+
N
5
×
10
cm
s
[75]
2
2
2
4
+
−
−6
0.5
3 −1
N4 + e → N2 (C) + N2
2 × 10 (Tg /Te ) cm s
[75]
+
N+
2.4 × 10−15 cm3 s−1
[75]
4 + N2 → N2 + 2N2
N2 + e− → N2 (A) + e−
f(E/N )
[17, 66]
N2 + e− → N2 (a0) + e−
f(E/N )
[17, 66]
N2 + e− → N2 (B) + e−
f(E/N )
[17, 66]
−
N2 (a0) + N2 (a0) → N+
1.0 × 10−11 cm3 s−1
[75]
4 + e
−
N2 (A) + N2 (a0) → N+
1.5 × 10−11 cm3 s−1
[75]
4 + e
N2 (a0) + N2 → 2 N2
2.0 × 10−13 cm3 s−1
[75]
N2 (A) + N2 (A) → N2 (B) + N2
7.7 × 10−11 cm3 s−1
[75]
N2 (A) + N2 (A) → N2 (C) + N2
3.0 × 10−10 cm3 s−1
[75]
N2 (A) + N → N2 + N
4.0 × 10−11 cm3 s−1
[75]
N2 (C) + N2 → N2 (a0) + N2
1.0 × 10−11 cm3 s−1
[75]
N + N + He → N2 + He
1.15 × 10−29 cm6 s−1
[29]
He∗2 + He → 2 He + He
1.0 × 104 cm3 s−1
[33]
−
2 He∗2 → He+
1.5 × 10−9 cm3 s−1
[33]
2 + 2 He + e
−
He∗2 + N2 → N+
3.0 × 10−11 cm3 s−1
[33]
2 + 2 He + e
−
He∗2 + e− → He+
9.75 × 10−10 × T e0.71 × e−3.4/T e cm3 s−1
[22]
2 + 2 e
He+ + He + e− → He + He∗
1.0 × 10−26 × (T g/T e)2.0 cm6 s−1
[22]
94
Reference
[29]
[29]
[29]
[29]
[29]
[17, 73]
[29]
[29]
[29]
[29]
[29]
[29]
[17, 55]
[17, 77]
[29]
[29]
[29]
[29]
Figure B.1 illustrates, in He with 1000 ppm of N2 and using the reaction scheme of tables B.1, B.2
and B.3, the reaction rates of electron-loss reactions and of Penning-type ionization reactions through
the internal energy of the excited species He∗ , He∗2 , N2 (A) (threshold energy 6.17 eV) and N2 (a0) (8.52
eV).
Figure B.1: Temporal evolution of the rates of the relevant electron-loss reactions (left), and of associative
and Penning ionization reactions (right), using the reaction scheme from tables B.1, B.2 and B.3, in the
99.9% He - 0.1% N2 plasma.
Figure B.2 shows a direct comparison of the species temporal evolution in the cases with and without
+
+
+
+
the charge transfer reactions He+
2 → N2 /N2 (B) and He → N2 (R13), (R13b), (R14), (R14b), (R21)
and (R22) for the 99.9% He - 0.1% N2 plasma case.
Figure B.2: Temporal evolution of the densities of the chemically relevant species using the reaction
scheme from tables B.1 and B.2 (left), and if charge transfer reactions (R13), (R14), (R21) and (R22)
are removed (right), in the 99.9% He - 0.1% N2 plasma.
In figure B.3, the zero-D results for the temporal evolution of the densities of the most relevant species
are represented including and removing the 3-body reactions from the reaction scheme, in the 99.9% He
- 0.1% N2 plasma case.
Figure B.4 represents the evolution of the densities of the most chemically relevant species in the
cases of 100 ppm of N2 in the He-N2 gas, which can be compared with results from figures IV.5 and
IV.9 for 1000 and 10 ppm of N2 , respectively. The conclusions to take from this figure are the same that
95
Figure B.3: Temporal evolution of the densities of the chemically relevant species using the reaction
scheme from tables B.1 and B.2 (left) and if 3-body reactions (R8), (R8b), (R14), (R14b) and (R22) are
removed (right), in the 99.9% He - 0.1% N2 plasma.
were taken in section IV.2.2 for the 10 ppm case. It is a clearly intermediary case between 10 ppm and
1000 ppm where [N2 ] defines both the electron-impact N2 ionization rates and the Penning ionization
rates and their importance is noticed. In figure B.4 we can also see the results for the case when the N2
concentration is the triple of what it was for 10 000 ppm of N2 in figure IV.9. For such high [N2 ], the gas
is clearly less ionized by the E field than for lower [N2 ]. The analysis is the same as the one presented in
section IV.2.2 for the 1% N2 case, except the ionization rate is much lower in the 3% case, thanks to the
electron-impact reactions and to the quicker decrease of the He species created during the discharge.
Figure B.4: Temporal evolution of the densities of the most chemically relevant species for the cases of
100 (left) and 30 000 (right) parts per million of N2 in the He-N2 plasma.
Zero-D results from figure IV.11 of the comparison of electron densities between the several He-N2
mixtures can be compared with those of figure B.5. This figure shows electron densities along the axial
positions in the tube at a given time of simulation (2 µs) and at the radial position where each electron
density is highest. We should take into account that the x-axis scale of figure B.5 is different from the
one of figure IV.11, since one is temporal and logarithmic and the other is spatial and linear. The results
are coherent, since the fronts have ne of the same order of magnitude and we clearly see that for the 1%
N2 case there is an increase in ne in the channel in post-discharge and a more advanced front.
96
Figure B.5: Axial evolution of the e− densities for the cases of 10, 100, 1000 and 10 000 parts per million
of N2 in the He-N2 discharge at t = 2 µs and at a fixed radial position, from the model presented in
section I.3.1.
To confirm that the electric field input in section IV.2.2 is a good estimation of the E in the discharge
front in the 2D model for Va = 6 kV for the studied N2 density cases, figure B.6 shows the longitudinal
electric field along the axial positions in the tube at a given time of simulation (2µs) and at the radial
position where each E is highest. The first conclusion to take from figure B.6 is that E in the front has
values between 11 and 12 kV/cm (44-49 Td) for the four orders of magnitude of N2 concentration.
Figure B.6: Axial evolution of the longitudinal E field for the cases of 10, 100, 1000 and 10 000 parts per
million of N2 in the He-N2 discharge at t = 2 µs and at fixed radial position, from the model presented
in section I.3.1.
97
98