Ionization in atmospheric-pressure Helium plasma jets€¦ · making a four-month internship in the...

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 GuerraProf. Luís Paulo da Mota Capitão Lemos Alves

Examination Committee

Chairperson: Prof. Maria Teresa Haderer de la Peña StadlerSupervisor: Prof. Vasco António Dinis Leitão Guerra

Members of the Committee: Dr. Olivier GuaitellaDr. Nuno Rombert Pinhão

June 2015

Acknowledgments

Firstly, I would like to thank my supervisors Profs. Vasco Guerra and Luıs Lemos Alves, for the oppor-

tunity 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 envi-

ronment for me. A special mention goes to my coworkers in the GEDG, Adriana Annusova 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 (Energetique Moleculaire et Macroscopique, Combustion)

research laboratory at Ecole Centrale Paris. I am particularly grateful to my supervisors in the EM2C,

Anne Bourdon and Francois Pechereau, 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 Ecole 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.

iii

Este trabalho foi financiado por uma Bolsa de Iniciacao Cientıfica no ambito do

projeto/instituicao de I&D, Incentivo-Descargas em Gases, 4467-Inc/3

Incentivo/FIS/LA0010/2014, financiado por fundos nacionais atraves da portuguesa

Fundacao para a Ciencia 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

Fundacao para a Ciencia e Tecnologia FCT/MEC (PIDDAC).

v

Resumo

Esta tese de mestrado e o resultado do trabalho desenvolvido no laboratorio EM2C (Energetique Moleculaire

et Macroscopique, Combustion) na Ecole Centrale Paris e no GEDG do IPFN no Instituto Superior

Tecnico - Universidade de Lisboa. Por isso beneficia das competencias de ambos os laboratorios e do

trabalho com diferentes equipas de investigacao, o que providencia um contexto unico de temas de inves-

tigacao, modelos e ferramentas de trabalho.

As aplicacoes medicinais levantam desafios para a comunidade dos plasmas. Sao necessarios plasmas

que funcionem no nosso ambiente e que se possam propagar em espaco aberto, a pressao atmosferica e

a temperatura ambiente. Jactos de plasma, onde uma descarga e guiada por um fluxo gasoso em tubos

e no ar, sao apropriados para estas aplicacoes. O gas mais usado e o Helio, misturado com pequenas

quantidades de Azoto e Oxigenio. A optimizacao das aplicacoes e a compreensao dos processos que

ocorrem nestes plasmas dependem da cinetica das especies no plasma.

O trabalho desenvolvido para esta tese inclui o desenvolvimento da ferramenta numerica IST-LoKI

para o estudo da cinetica dos plasmas. A cinetica electronica em misturas He-N2-O2 e estudada,

mostrando-se os efeitos do Helio em estado excitado, da mistura dos gases moleculares ao Helio e da

electronegatividade do Oxigenio. Sao comparados modelos cineticos para Helio puro em estado esta-

cionario e sao obtidos resultados para diferentes condicoes de descarga. Finalmente, o efeito de um

campo electrico transitorio num plasma He-N2 e simulado usando a ferramenta ZDPlasKin.

Palavras-chave: Jactos de plasma, Helio, Cinetica electronica, Cinetica quımica, Ionizacao,

Aplicacoes medicinais dos plasmas

vii

Abstract

This master thesis is the result of work developed in both the EM2C (Energetique Moleculaire et Macro-

scopique, Combustion) laboratory at Ecole Centrale Paris and the GEDG group of IPFN at Instituto

Superior Tecnico - Universidade de Lisboa. Therefore, it benefits from the competences of both labora-

tories 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

ix

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

I.2 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

I.3 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

I.4 Plan of the thesis and original contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 19

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

IIIKinetic 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

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(11S) and He(23S) . . . . . . . . . . . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xv

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(23S) and He(21S) relative densities as f(ne) from [12] and from IST-LoKI using the

reaction scheme from table III.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

III.6 He(23P ) 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


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


III.13Rates of creation and destruction of He(23S) as f(ne) from IST-LoKI using the reaction


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

IV.4 Rates of production of N+2 (B2Σu) and rates of N2(C → B) and N+

2 (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 . . . . . . . 67

xvi

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(C3Πu) 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(11S) 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

A.8 Swarm parameters for different mixtures of He(11S) and He(23S) . . . . . . . . . . . . . . 89

A.9 Swarm parameters for different mixtures of He(11S), He(23S) and He(21S) . . . . . . . . 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xvii

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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

xviii

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

Chapter IIntroduction and state of the art

I.1 Introduction

I.1.1 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/(KBT0) = 1012 − 1014 cm−3) for use in various applications. In

this case there is a higher mean free path of electrons (〈λ〉 = (n0〈σe0〉)−1) and a lower electron-neutral

collision frequency (νe0 = n0〈σe0(ve)ve〉) 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 ∼ 100sµm, 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∂t

+∇.(ns ~vs) = Ss − Ls (I.1)

~js = ns ~vs = nsµs ~E −Ds∇ns (I.2)

∇. ~E =

∑i qiniε

(I.3)

Ss, Ls =∑j

∏i=1,2,3

ni

Kj (I.4)

2

Ds =kBTsmsνs0

, µs =qs

msνs0(I.5)

Kj , Ds, µs = f(E/N, gas mix) (I.6)

In these equations, ns is the number density of species s, ~vs is its velocity and ~js 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

mass, qs its charge and νs0 its electron-neutral collision frequency. The electric field is ~E = −∇V , where

V is the electric potential and ε 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, thin-

film 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, radio-

frequency 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 simulatedring-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

(H2O2), for teeth bleaching in dental clinics [7]. It was demonstrated that a tooth exposed to a He jet

with H2O2 becomes brighter when compared with a tooth treated with H2O2 alone. Acceleration of tooth

bleaching by the jet results from the enhancement of OH radical production due to the H2O2 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 50oC, 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 burn-

wounds 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 anti-

tumor 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 mouseon 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 treat-

ment 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 non-

destructive 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 Gas discharge kinetics

I.2.1 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, cou-

pled 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(ε) may be hard to obtain experimentally and sim-

ulations 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, ~ve, t)

∂t+ ~ve. ~∇rfe(~r, ~ve, t)−

e

me

~E(~r, t). ~∇vfe(~r, ~ve, t) =

(∂fe∂t

)coll

(~r, ~ve, t) (I.7)

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 electron-

impact reaction, described by cross section σ(ve), the rate coefficient ks writes:

ks(E/N) = 〈σ(ve)ve〉 =

∫σ(ve)vefe(ve, E/N)4πv2edve (I.8)

For collisions between neutral species, the rate coefficients usually follow the Arrhenius expression:

ks(T0) = BsTαs0 exp

(− EaskBT0

), (I.9)

where Eas is the activation energy per molecule, kBT0 is the average kinetic energy of the neutral species,

BsTαs0 is the total frequency of collisions and e−Eas/(kBT0) 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 electron-

impact 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 high-

9

pressure discharges are photo-ionization and associative and Penning ionization processes, in which ex-

cited 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 com-

ponents

In the context of He applications (subsection I.1.3), the starting point must be the development of state-

of-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+1l) excited states (where n, l and s are

the principal, the orbital and the spin quantum numbers, respectively), ground-state atomic ions He+,

ground-state molecular ions He+2 (23SΣ+u ) and He∗2 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(23S), He(21S), He∗2, He+, He+2 ), which have long lifetimes τ (e. g., the metastable

states He(23S) and He(21S) have τ ∼ 8400 s and 2 × 10−2 s, respectively, while the radiative states

He(23P ) and He(21P ) 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(23S)). The coefficients of heavy particle reactions are either constant or gas temperature dependent.

The metastable states He(23S) and He(21S) and the radiative states He(23P ) and He(21P ), 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 H2O2 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)/N+

2 (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 two-

dimensional 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 , N+2 ), as well as three excited

species (He(23S)+He(21S), 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 partic-

ularly 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(23S)

12

excited state can have importance in He-based mixtures as a metastable state by its own, not together

with He(21S). References [38–40] follow the He(23S) 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 applica-

tions 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−, O−2 and O−3 and metastable excited states O∗ and O∗2. 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 de-

creases 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 H2O, instead

of the more simple dry air model using 80% N2 - 20% O2. The work adopts the set of reactions for

He-H2O 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

active species He+, O+2 , N+

2 , N, O, He∗, N∗2, N(2D), O(1D), O2(a1∆) and O2(b1Σ), 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(1D) + H2O

→ OH + OH; and NO created by N(2D) + 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 atten-

tion 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 momentum-

transfer 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(23S) 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 atatmospheric-pressure, calculated for a cylindrical container with R = 1 mm and L = 2.3 cm and anapplied 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 es-

sential 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 (Energetique Moleculaire et

Macroscopique, Combustion) laboratory at Ecole Centrale Paris and the Gas Discharges and Gaseous

Electronics (GEDG) group of IPFN (Instituto de Plasmas e Fusao Nuclear) at Instituto Superior Tecnico

- 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 Ecole 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 (εr = 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 = 109cm−3 and a constant voltage Ua = 6

kV, applied to the anode ring for t ≥ 0. The time-step varies between 10−11s and 10−12s.

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 co-

efficients 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 ob-

tained 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 BOL-

SIG+ [17, 55] with fitted formulas. Positive ion mobilities vary less than 20% with E/N and are taken

as constant average values: µN+2

= 20, µHe+ = 10.0, µHe+2= 17.5 cm2V−1s−1. Diffusion coefficients for

ions are found using the Einstein relation Di

µi=

kBTg

qiwith 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 densityduring 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 Tecnico

(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 zero-

dimensional (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 ~ve 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

f0e (ve) and the anisotropic component f1e (ve) of the EEDF as:

1

v2e

d

dve[v2e(g0E + g0e−0)] = 0 (I.10)

g0E = − 1

3νc

ν2cν2c + ω2

(eE

me

)2df0edve

> 0; g0e−0 = −me

Mveνc

(f0e +

kBTgmeve

df0edve

)< 0 (I.11)

~g = (g0E + g0e−0)~ev; f1e = − 1

νc + jω

eE

meνc

df0edve

(I.12)

Here, g0E ~ev is a radial outward vector in velocity space, since it represents the heating due to the electric

field, while g0e−0 ~ev is directed inward, representing the cooling of electrons due to elastic collisions and

which can be divided as g0e−0 = g0el + g0exc + g0ion− g0sup, with respect to elastic, excitation, ionization and

superelastic collisions. The product 4πv2eg = 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 NR. Here, NR

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 zero-

D 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 IIElectron kinetics in Helium-based plasmas

II.1 General formulation for the numerical solution of the elec-

tron 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(11S), 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(23S) (g = 3) and He(21S) (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 4πv2e to obtain the flux, then

by the electron energyv2eme

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 Validation of collisional data

II.2.1 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(11S) collision cross sections from [12] and there is no effect of su-

perelastic collisions. These cross sections are for elastic collisions, inelastic collisions towards all the states

He(n2s+1l) 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 ground-

state He, of excitation collisions e− + He (11S) → e− + He (23S) with energy threshold 19.82 eV and of

ionization collisions e− + He (11S) → 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

results of a Maxwellian distribution in thermal equilibrium (fMB = C×e−u(eV )Te(eV ) , Te = 2

3 u), that uses the

average energy u 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 thresh-

old 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-stateHelium 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:

µe ×N = −√

2e/me

3

∫ +∞

0

u

σeff

dfedu

du (II.2)

De ×N =

√2e/me

3

∫ +∞

0

u

σefffedu (II.3)

vde = µe × E;uK = De/µe; α/N = Kion/vde (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 =PJE/N

; µe =vdeE

(II.5)

uK = De/µe; α/N = Kion/vde (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+1l, 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(11S) as function of E/N , obtained from several models andexperiments: reduced electronic diffusion coefficient on the left and reduced electronic mobility on theright.

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+1l, 4 ≤ n ≤ 7, present in [12], would not provoke a

24

Figure II.3: Swarm parameters for pure He(11S) 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/Nfrom 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

( αN −ηN = KN2+×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

compare the results from LoKI, for both the ionization coefficient alone ( αN = KN2+×0.8+KO2+×0.2vde

) 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 airfrom LoKI (LoKI effective), and reduced Townsend ionization coefficient (not effective) for dry air fromLoKI (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(23S) and He(21S), 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(23S)

(statistical weight g = 3) and He(21S) (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(11S) and

excited states He(23S) and He(21S). 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:

σij(u) =σji (u+ ∆uji )(u+ ∆uji )

u

gigj

(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(21S) → He(21P ) and the He(23S) → He(23P ), 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(11S) cross sections and

the addition of only one superelastic process from an excited state towards the ground-state (He(23S) →

He(11S)), for different relative densities of He(11S) and He(23S). On the right-hand side of figure II.6,

all the inelastic processes between electrons and He(23S) are included, in addition to those previously

considered. The values of He(23S) 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(11S) andHe(23S). On the left, only e− + He(11S) processes and superelastic He(23S) → He(11S) process areincluded. In the legends He(x, y, z), x, y and z are the relative densities of He(11S), He(23S) andHe(21S). On the right, all electron-impact processes from He(11S) and He(23S) 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(23S)], 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(11S) and He(23S). 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(21S) densities instead of He(23S) 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(21S)

→ He(23S), He(23S) → He(11S) and He(21S) → He(11S) 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(21S), 4.77 eV for collisions with

He(23S) 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(23S)]=[He(21S)]. For these E/Ns, 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) andseveral mixtures of He(11S), He(23S) and He(21S) with [He(23S)] = [He(21S)].

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(23S) and He(21S), 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(23S) and He(21S), 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 ioniza-

tion coefficient α/N for varying ground-state He and He(23S) 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) asfunction of E/N for different mixtures of He(11S) and He(23S).

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(23S) (10−4), stepwise ionization is relevant until ∼100 Td, whereas for low values

of He(23S) 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(23S), He(21S) 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 ki-

netics

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(11S) and N2

(left) and of He(11S) 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(23S) and He(21S) 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(11S) anddry air (80% N2 - 20% O2) (left) and for E/N = 1 Td and several mixtures of He(11S), He(23S), He(21S)and O2, with He(23S) and He(21S) 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(23S) and He(21S). 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] +KN2+[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

(KN2+× [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 contributionof 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 electron-

impact 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 electron-impact ionization coefficient, as function of the relative density of O2, for three values of E/N : 5 Td, 20Td 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 Heon 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(23S) and He(21S), 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 ground-

state 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 elec-

trons 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 dis-

tribution 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 ion-ization on the right, as function of the relative density of dry-air (80% N2 - 20% O2), for three values ofE/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

Chapter IIIKinetic 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). IST-

LoKI 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 NR, 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(23S) and He(21S), 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(23S)), 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) ≡

He(n2s+1l) excited states, ground-state atomic ions He+, ground-state molecular ions He+2 (23SΣ+u ) 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(11S), He(23S), He(21S), He(23P ) and He(21P ). 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(23S), He(21S), 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(23S) and He(21S), associative ionization

He(n > 2, l, s) + He → He+2 + e− and quantum number exchange reactions He(n > 2, l, s) + He →

He(n > 2, l′, 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

Table III.1: Reaction scheme for He at atmospheric-pressure, reduced from [12].Nr Process Reaction Rate coefficient

R1-4 e-impact exc. He(11S) + e− → He(2) + e− EEDF (E/N)R5-7 He(23S) + e− → He(21S/23P/21P ) + e− EEDF (E/N)R8-9 He(21S) + e− → He(23P/21P ) + e− EEDF (E/N)R10 e-impact de-exc. He(23S) + e− → He(11S) + e− EEDF (E/N)

R11-12 He(21S) + e− → He(23S/11S) + e− EEDF (E/N)R13-15 He(23P ) + e− → He(21S/23S/11S) + e− EEDF (E/N)R16-18 He(21P ) + e− → He(21S/23S/11S) + e− EEDF (E/N)R19-21 e-impact ion. He(11S/23S/21S) + e− → He+ + 2 e− EEDF (E/N)R22-25 Diss. recomb. He+2 + e− → He(11S) + He(2) EEDF (E/N)

R26 Excimer ion. He∗2 + e− → He+2 + 2 e− 9.75× 10−10T 0.71e e−3.4/Te cm3 s−1

R27 3-body recomb. He+ + 2 e− → He(23S) + e− 1.1× 10−14T 2.3g /T 4.5

e cm6 s−1

R28 He+2 + 2 e− → He(23S) + He(11S) + e− 0.5× 1.1× 10−14T 2.3g /T 4.5

e cm6 s−1

R29 He+2 + 2 e− → He∗2 + e− 0.5× 1.1× 10−14T 2.3g /T 4.5

e cm6 s−1

R30-39 Penning ion. He(2) +He(2) → He+ + He(11S) + e− 0.3× 2.9× 10−9(Tg(K)300

)−1.86cm3 s−1

R40-49 He(2) +He(2) → He+2 + e− 0.7× 2.9× 10−9(Tg(K)300

)−1.86cm3 s−1

R50 Charge transfer He+ + 2 He(11S) → He+2 + He(11S) 1.4× 10−31(Tg(K)300

)−0.6cm6 s−1

R51 He+2 + He(11S) → He+ + 2 He(11S) 1.4×10−6

Tg(K)−0.6 e−28100/Tg(K) cm3 s−1

R52 Association He(23S) + 2 He(11S) → He∗2 + He(11S) 1.5× 10−34 cm6 s−1

R53 He(23P ) + 2 He(11S) → He∗2 + He(11S) 1.6× 10−32 cm6 s−1

R54 Dissociation He∗2 + He(11S) → He(23P ) + 2 He(11S) 3.6× 10−14 cm3 s−1

R55 He∗2 + e− → 2 He(11S) + e− 4.0× 10−9 cm3 s−1

R56 Radiative trans. He(23P ) → He(23S) + hν 1.018× 107 s−1

R57 He(21P ) → He(21S) + hν 0.180× 107 s−1

R58 He(21P ) → He(11S) + hν 47.728× 107 s−1

R59-60 Diff. to the wall He(23S/21S) DM = 8.922× 10−2Tg(K)1.5

p(Torr)cm2

s

R61 Diff. to the wall He∗2 D∗ = 7.102× 10−2Tg(K)1.5

p(Torr)cm2

s

R62 Diffusion He+ µa ×N = 2.68×10192.96×10−3T 0.5

g +3.11×10−2cm2

V s

Da ×N = µ× kBTg(K)qe

cm2

s

R63 Diffusion He+2 µm ×N = 2.6× 1019 cm2

V s

Dm ×N = µ× kBTg(K)qe

cm2

s

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(23S), 34.1% He(21S), 64.5% He(23P ) and 0.3% He(21P ). The electron-stabilized recombination

of He+ (R27) is considered to produce only He(23S) 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(23S) and

He(21S) and radiative states He(23P ) and He(21P ), with branching ratios 0.3 and 0.7 for the production

of He+ and He+2 , respectively. The rate coefficient adopted follows the expression A(Tg(K)/300)B , which

41

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(23S) and He(23P ) 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(23P ), 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(23P ) and He(21P ) 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(23S) and 2 × 10−2 s for He(21S)), 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 = kBTg/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(11S). The steady-state spatially averaged rate balance equation for a

given species j is written as

Sj 'Dj

Λ2nj (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

naDa,a + nmDa,m ' neDa,e (III.2)

Da,e =na(µaDe + µeDa) + nm(µmDe + µeDm)

naµa + nmµm + neµe(III.3)

42

Da,a =nm(µmDa + µaDm) + ne(µaDe + µeDa)


Da,m =na(µaDm + µmDa) + ne(µmDe + µeDm)


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 Eeff/N is also

represented, defined as

EeffN

=E

N

νc(ν2c + ω2)1/2

(III.6)

where ω = 2πf and νc is the total frequency of momentum-transfer electron-neutral collisions. Eeff/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 Eeff/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/(kBTg).

Figure III.1: Reduced electric field on the left and He ground-state density on the right, as function ofthe electron density, from [12] with and without electron-electron collisions and from IST-LoKI using thereaction 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 charged-

particle 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 IST-

LoKI. Conversely, the comparison of the densities of the main species in the plasma, He(11S), 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 theright, as function of the electron density, from [12] with and without electron-electron collisions and fromIST-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(23S) and He(21S), 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 stepwiseionization (right), normalized to the total creation rates of charged species, as function of the electrondensity, from [12] with and without electron-electron collisions and from IST-LoKI using the reactionscheme from table III.1.

The exclusion of species He(n > 2, l, s) from the CRM of table III.1 implies the exclusion of electron-

impact 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 (R30-

49) 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 throughdiffusion (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 fromtable 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(23S) and He(21S) 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(23P ) and He∗2.

Figure III.5: Helium atomic metastable densities He(23S) on the left and He(21S) on the right, normalizedto the total gas density, as function of the electron density, from [12] with and without electron-electroncollisions and from IST-LoKI using the reaction scheme from table III.1.

Figure III.6: Helium densities of atomic radiative species He(23P ) on the left and molecular excimer He∗2on the right, normalized to the total gas density, as function of the electron density, from [12] with andwithout 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(23S), He(23P ) and He∗2 are close to 10−7

47

(in absolute values ∼ 1011-1012 cm−3), while the densities of He(21S) and He(21P ) are close to 10−8 (∼

109-1011 cm−3). However, in LoKI the density of He(21S) is higher than that of He(21P ) 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(21S)], while in our model these densities keep one order of

magnitude apart.

In both models, He(23S) is destroyed by electron-impact excitation and stepwise ionization. The

He(23S) creation processes are also common. At low ne, radiative transition (R56) (∼60% of the He(23S)

creation rates), electron-impact de-excitation from He(23P ) (R14) (∼30%) and from He(21S) (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(23P ) is created by both electron-impact excitation from He(23S) (R6)

and by He+2 dissociative recombination (R24). However, (R6) contributes only with ∼2% for the He(23P )

creation in LoKI, while in [12] this contribution goes up to ∼ 20%, due to the difference in [He+2 ]. The

He(23P ) 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(23S) (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(21S) and He(21P )

are also similar in both models. At low ne, He(21S) is created by excitation from the ground-state

(R2) (∼20%) and from He(23S) (R5) (∼60%) and by de-excitation from He(23P ) (R13) (∼13%) and is

destroyed by electron-impact excitation towards He(23P ) (R8) (∼5%) and towards He(21P ) (R9) (∼90%)

and by electron-impact de-excitation towards He(23S) (R17) (∼5%). At high ne, the rates for He(21S)

creation and destruction slightly change, but the dominant processes remain the same. Finally, He(21P )

is created at low ne by electron-impact excitation from He(11S) (R4) (∼15%), from He(23S) (R7) (∼15%)

and from He(21S) (R9) (∼65%) and destroyed mostly by the radiative transition towards the ground-

state (R58) (∼99%). At high ne, the only significant change in He(21P ) rates, is the increase in the rate

of electron-impact de-excitation towards He(21S) (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: uMAX = 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 functionof 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(11S)] 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.

Figure III.8: Helium ion species He+ and He+2 densities (left) and excited species He(23S), He(21S),He(23P ), He(21P ) and He∗2 densities (right), as function of time, calculated by IST-LoKI using thereaction 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+2 and e− densities (left), and excited species He(23S), He(21S),He(23P ), He(21P ) and He∗2 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(23S), the radiative state He(23P ) and the excimer He∗2 can have important densities, overcoming

ne in some occasions. He(21P ) 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(23S), He(23P ) and He∗2 were approximately 1 order of magnitude lower

and the densities of He(21S) and He(21P ) 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(23S) and He(21S) 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.

Excimer ionization (R26) appears to be important in He+2 creation, specially at high ne = 1012−1014

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(23S)],

51

Figure III.10: Rates of creation (left) and destruction (right) of He+2 , discriminated by reaction, asfunction of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. Thenumbers of reactions refer to table III.1.

[He(21S)] and [He(23P )] are relevant at low ne and reactions (R40,42,47), that depend on [He(23S)] and

[He(23P )] are relevant at high ne, while reactions (R43,46,48,49), that depend on [He(21P )] appear to

have irrelevant rates, which reinforces the possibility of removing He(21P ) from a more reduced model.

Comparing with the He+2 creation rates of the previous section for ne = 1013 cm−3, where Tg = 1536

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(23S) and He(21S) 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, asfunction of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. Thenumbers 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(23S) and

He(23P ) in rates (R30,31,32,37) and reinforcing the irrelevance of He(21P ) in rates (R33,36,38,39). Once

again, comparing the He+ creation rates in both models (Tg = 300 K and 1536 K) evidences that charge-

creation 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(23S) balance.

53

Figure III.12: Rates of creation (left) and destruction (right) of electrons, discriminated by reaction, asfunction of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. Thenumbers of reactions refer to table III.1.

Figure III.13: Rates of creation (left) and destruction (right) of He(23S), discriminated by reaction, asfunction of the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. Thenumbers 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(11S)]. The electron-impact de-excitation

from He(21S) (R11) is not irrelevant for He(23S) creation and actually it is essential for the equilibrium

between the He excited-states densities. The transitions to He(23S) from He(23P ) 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(23P ) has a more relevant role. The de-excitation from He(21P ) (R17) is

not relevant for He(23S) creation, even at high ne, which contributes to reinforce the idea that a reduced

scheme can dismiss He(21P ). The dissociative recombination of He+2 , that creates He(23S) (R22), is

not important for its creation but maintains its relevance for charge loss. Finally, electron-stabilized

recombination reactions that produce He(23S) (R27-28) are again irrelevant for the species balances.

From the right figure III.13, we notice that stepwise excitation reactions from He(23S) to higher

states (R5-7), and towards He(23P ) (R6) in particular, have important rates for the He(23S) 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(23S) 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(11S) density, assumes an important role in

He(23S) loss at low ne, below 1013 cm−3. Even though the diffusion of He(23S) (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(23S) towards He(11S) (R10) and the Penning ionization reactions with He(23S) (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(23S) 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 functionof the electron density, calculated by IST-LoKI using the reaction scheme from table III.1. The numbersof 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(23S) and He(23P ) (R52-53) have total influence on

the He∗2 creation. The association with He(23S) (R52) is dominant at low ne and relevant at high ne,

while the reaction with He(23P ) (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(11S) 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(11S) and He(23P ). 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(11S), 5 times higher when Tg = 300 K.

The rates describing the balances of the other He atomic excited species He(21S), He(23P ) and

He(21P ) are not represented here. He(21S) 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(23P ) 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(21P ) are the density exchanges with He(11S) and

He(21S) 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(21P ) 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 IVZero-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 correspond-

ing 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 simu-

lation 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 po-

57

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 shift = 6 ns. The

slope rate is λ = 8/trise. Its formulation is given by:

E = Emin + (Emax − Emin)× (1

1 + e−λ×(t−shift)+

1

1 + e+λ×(t−shift−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

ions are considered in this scheme (He+, He+2 , N+2 ), as well as three excited species (He(23S)+He(21S),

N2(C), N+2 (B)). Reactions (R15) to (R18) are taken into account to compute the time evolution of the

N2(C3Πu) 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 N+2 , 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(23S) and He(21S), also arises during the electric field application but stays longer in post-

discharge. 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 plasmawith 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

front, [He+], [N+2 ], [N2(C3Πu)] 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 → N+2 (R13) and (R14). [He+2 ] 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 N+

2 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

post-discharge is clearly N+2 with densities above 1012 cm−3. These results confirm that we can obtain a

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 atsimulation 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(23S) and He(21S) but only as He(23S), which reaches values 2 to 3 orders of magnitude greater

than He(21S). 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(23S) 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(23S), since the other loss processes consist of ionizations. It is

therefore a process that removes He(23S) 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.

Reference [29] introduces the excited ion N+2 (B2Σu), 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

of production of N+2 (B). Electron-impact ionization towards N+

2 (B) shows a weak rate (max 1018 cm−3

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) → N+

2 (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.

Figure IV.4: Temporal evolution of the rates of production of N+2 (B2Σu) (left), and N2(C → B) and

N+2 (B → X) light emission rates (right), in the 99.9% He - 0.1% N2 plasma using the reaction scheme

from tables B.1 and B.2.

Charge transfer reactions He+ → N+2 (R21) and (R22) are very important in He+ loss, decreasing

[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 He+2 → N+2 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+ → N+

2 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 fromtables 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(21S) 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+ → N+

2 but is lower

between 50 ns and 500 ns, due to the share of He+2 → N+2 charge transfer and Penning ionization with

the new species N+2 (B). Since N+

2 (B) is lost through de-excitation to N+2 , by 500 ns we obtain ne = [N+

2 ],

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

of He+ → N+2 /N+ + N (R27-28). In fact, electron-impact ionization rates of N (∼ 1017 cm−3 s−1) and

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(a3Σ+u ), called He∗2, 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

excimer He2(a3Σ+u ), the metastables N2(A3Σ+

u ), N2(a′1Σ−u ) and the ion N+4 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(a′) (threshold 8.52 eV).

Figure IV.6: Temporal evolution of the densities of the relevant charged species (left), and of somechemically 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(23S) 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(a′) 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 He∗2 and the nitrogen

metastables N2(A) and N2(a′) 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(23S) 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 → N+2 /N+

2 (B) and He+ → N+2 (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 He+2 is not subject to charge transfer

towards N+2 , its density stays almost constant starting from 40 ns. In this case, N+

2 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 ] + [N+2 ]. The difference is higher when He+ and He+2 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 /N+

2 (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 reactionscheme 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(23S)] decreases slower if there is no three-body

reaction. Instead of [He(23S)] 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 µssimulation time, using the reaction scheme from tables B.1 and B.2 with Penning reactions (left), andwithout 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 concen-

trations 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 electron-

impact 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 N+2 will still act slowly, being the

most responsible reactions for the N+2 increase starting at t = 20 ns. He+2 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 of10 (left) and 10 000 (right) parts per million of N2 in the He-N2 plasma, using the reaction scheme fromtables 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 N+2 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(23S) 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(C3Πu) 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.1and 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 ofthe electron/ion densities (left), and of the creation rates by Penning ionization (R7) (right), using thereaction 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, 1000and 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 + e− → 2 N (R11), for all the

He-N2 mixtures considered in this chapter. At that point, since ne = [N+2 ], the electron loss equation is

formulated as:

∂ne∂t

= −Krecombinationn2e (IV.2)

ne(t) =1

Krect+ 1n′e

' 1

Krectfor t ≥ 1

Krecn′e(IV.3)

Where n′e 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 tMAX , equation (IV.3) is solved, providing the result in eq. (IV.4). tMAX 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 tMAX follows that result and that in the same mixture, for different pre-ionization levels chosen,

there is a different tMAX proper for pulse application. Therefore, a frequency can be defined so that a

pre-ionization is fixed along a whole cycle of pulses.

ne(tMAX) = ne0 <=> tMAX =1

Krecne0(IV.4)

f = Krecne0, with Krec = 1011cm3s−1 (IV.5)

Figure IV.13: Temporal evolution of the electron density, using the reaction scheme from tables B.1and 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

electrons and of N+2 should be assumed and if one intends to use a 1 kHz pulser, a 1010 cm−3 of ne0 should

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

Chapter VConclusions 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 electron-

impact 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(23S) and He(21S) 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(21P ) 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 IST-

LoKI 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 (Energetique Moleculaire et Macroscopique, Combus-

tion) 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 mix-

tures 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(23S), 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 N+

2 (B) and N+4 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(23S) may be considered.

Coupling zero-D and 2D model results allowed us to understand the role of species evolution and re-

actions 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 explana-

tion 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(23S) → 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+2 + e− → N + N in late post-discharge that makes kinetics in

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 AElectron 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(11S) → e− + He(23S) and of ionization collision e− + He(11S) → 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(11S) 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-stateHelium in logarithmic scale for different input reduced electric fields 1 Td (uMAX=60 eV, N=6000), 10Td (uMAX=125 eV, N=7500), 50 Td (uMAX=250 eV, N=7500) and 100 Td (uMAX=550 eV, N=7500).

Figure A.3: Results for the calculated and Maxwellian EEDFs when the gas is completely ground-stateHelium in logarithmic scale for different input reduced electric fields 250 Td (uMAX = 1000 eV, N =7500), 500 Td (uMAX = 1000 eV, N = 7500) and 1000 Td (uMAX = 1000 eV, N = 7500).

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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 ex-periments: reduced electronic mobility, reduced electronic diffusion coefficient and electron characteristicenergy.

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

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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 ofE/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(21S)] and [He(23S)]=[He(21S)] on the

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EEDF for E/N=1 Td.

Figure A.7: Results for the calculated EEDFs in logarithmic scale for E/N=1 Td and several mixturesof He(11S) and He(21S) (left) and He(11S), He(23S) and He(21S) (right).

Figure A.8 presents the swarm parameters De ×N , µe ×N and uK for varying ground-state He and

He(23S) 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(11S) and He(23S) as function of E/N : reducedelectronic diffusion coefficient De × N , reduced electronic mobility µe × N and electron characteristicenergy uK .

Results of swarm parameters De×N , µe×N , uK , α/N and total ionization coefficient Kion obtained

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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(23S), He(21S) 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 electronicdiffusion coefficient De×N , reduced electronic mobility µe×N , reduced Townsend ionization coefficientα/N , ionization coefficient Kion and electron characteristic energy uK .

As far as the reduced electronic diffusion coefficient, the reduced electronic mobility and the charac-

teristic 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, respec-

tively, 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(11S) 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(11S) 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(11S) 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(23S) and He(21S).

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(11S), He(23S), He(21S) and O2, with He(23S) and He(21S)relative densities 10−4.

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Appendix BZero-dimensional simulation of an

atmospheric-pressure He-N2 tube streamer

Table B.1: Original kinetic scheme for a He plasma with 1000 ppm of N2 from [26].Nr Process Reaction Rate coefficient ReferenceR1 Direct ionization He + e− → He+ + 2 e− f (E/N) [17, 66]R2 N2 + e− → N+

2 + 2 e− f (E/N) [17, 66]R3 Direct excitation He + e− → He∗(23S/21S) + e− f (E/N) [17, 66]R4 De-excitation He∗ + e− → He + e− 2.9 × 10−9 cm3 s−1 [29]R5 Associative ionization He∗ + He∗ → He+ + He + e− 4.35 × 10−10 cm3 s−1 [29]R6 He∗ + He∗ → He+2 + e− 1.015× 10−9 cm3 s−1 [29]R7 Penning ionization He∗ + N2 → He + N+

2 + e− 7.6× 10−11 cm3 s−1 [29]R8 3-body Penning He∗ + N2 + He → 2 He + N+

2 + e− 3.3× 10−30 cm6 s−1 [29]

R9 Recombination He+ + e− + e− → He + e− 8.0× 10−20(Tg

Te)4 cm6 s−1 [30]

R10 Dissociative recomb. He+2 + e− → He + He 9.0× 10−9 cm3 s−1 [29]R11 N+

2 + e− → N + N 9.0× 10−9 cm3 s−1 [29]R12 Ion conversion He+ + 2 He → He+2 + He 1.5× 10−31 cm6 s−1 [29]R13 Charge transfer He+2 + N2 → N+

2 +2 He 1.1× 10−9 cm3 s−1 [29]R14 He+2 + N2 + He → N+

2 +3 He 1.36× 10−29 cm6 s−1 [29]R15 Direct excitation N2 + e− → N2(C3Πu) + e− f (E/N) [17, 66]R16 De-excitation N2(C3Πu) + N2 → N2 + N2 8.0× 10−11 cm3 s−1 [76]R17 N2(C3Πu) + He → N2 + He 1.0× 10−12 cm3 s−1 [76]R18 Radiative decay N2(C3Πu) → N2(B3Πg) + hν 2.4 ×107 s−1 [76]

93

Table B.2: List of proposed reactions to add to those of table B.1.N br Process Reaction Rate coefficient Reference

R19 Dissociative recomb. He+2 + e− → He(23S) + He 5.0× 10−9 cm3 s−1 [29]R20 Association He(23S) + 2 He → He∗2 + He 1.9× 10−34 cm6 s−1 [29]R21 Charge transfer He+ + N2 → He + N+

2 1.2× 10−9 cm3 s−1 [29]R22 He+ + N2 + He → 2 He + N+

2 2.2× 10−29 cm6 s−1 [29]R23 N Association N + N + He → N2 + He 1.15× 10−29 cm6 s−1 [29]R2b Direct ionization N2 + e− → N+

2 (B) + 2 e− f (E/N) [17, 73]R7b Penning ionization He(23S) + N2 → He + N+

2 (B) + e− 0.5× 7.6× 10−11 cm3 s−1 [29]R8b 3-body Penning He∗ + N2 + He → 2 He + N+

2 (B) + e− 0.5× 3.3× 10−30 cm6 s−1 [29]R11b Dissociative recomb. N+

2 (B) + e− → N + N 0.5× 2× 10−7 cm3 s−1 [29]R13b Charge transfer He+2 + N2 → N+

2 (B) +2 He 0.25× 1.1× 10−9 cm3 s−1 [29]R14b He+2 + N2 + He → N+

2 (B) + 3 He 0.25× 1.36× 10−29 cm6 s−1 [29]R24 Radiative decay N+

2 (B) → N+2 (X) + hν 1.5× 107 s−1 [29]

R25 Direct ionization N + e− → N+ + 2 e− f (E/N) [17, 55]R26 Dissoc. ionization N2 + e− → N+ + N + 2 e− f (E/N) [17, 77]R27 Charge transfer He+ + N2 → N+ + N + He 0.5× 1.2× 10−9 cm3 s−1 [29]R28 He+ + N2 + He → N+ + N + 2 He 0.5× 2.2× 10−29 cm6 s−1 [29]R29 Penning ionization He(23S) + N → N+ + e− + He 1.6× 10−10 cm3 s−1 [29]R30 Recombination N+ + e− → N 5.0× 10−9 cm3 s−1 [29]

Table B.3: List of proposed reactions to add to those of tables B.1 and B.2.N br Reaction Rate coefficient Reference

R31 He+2 + N2 → N+2 (B) + He∗2 0.75× 1.1× 10−9 cm3 s−1 [29, 33]

R32 N+2 + 2 N2 → N+

4 + N2 5× 10−29 cm6 s−1 [75]R33 N+

4 + e− → N2(C) + N2 2× 10−6 (Tg/Te)0.5 cm3 s−1 [75]

R34 N+4 + N2 → N+

2 + 2N2 2.4× 10−15 cm3 s−1 [75]R35 N2 + e− → N2(A) + e− f(E/N) [17, 66]R36 N2 + e− → N2(a′) + e− f(E/N) [17, 66]R37 N2 + e− → N2(B) + e− f(E/N) [17, 66]R38 N2(a′) + N2(a′) → N+

4 + e− 1.0× 10−11 cm3 s−1 [75]R39 N2(A) + N2(a′) → N+

4 + e− 1.5× 10−11 cm3 s−1 [75]R40 N2(a′) + N2 → 2 N2 2.0× 10−13 cm3 s−1 [75]R41 N2(A) + N2(A) → N2(B) + N2 7.7× 10−11 cm3 s−1 [75]R42 N2(A) + N2(A) → N2(C) + N2 3.0× 10−10 cm3 s−1 [75]R43 N2(A) + N → N2 + N 4.0× 10−11 cm3 s−1 [75]R44 N2(C) + N2 → N2(a′) + N2 1.0× 10−11 cm3 s−1 [75]R45 N + N + He → N2 + He 1.15× 10−29 cm6 s−1 [29]R46 He∗2 + He → 2 He + He 1.0× 104 cm3 s−1 [33]R47 2 He∗2 → He+2 + 2 He + e− 1.5× 10−9 cm3 s−1 [33]R48 He∗2 + N2 → N+

2 + 2 He + e− 3.0× 10−11 cm3 s−1 [33]R49 He∗2 + e− → He+2 + 2 e− 9.75× 10−10 × Te0.71 × e−3.4/Te cm3 s−1 [22]R50 He+ + He + e− → He + He∗ 1.0× 10−26 × (Tg/Te)2.0 cm6 s−1 [22]

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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(a′) (8.52

eV).

Figure B.1: Temporal evolution of the rates of the relevant electron-loss reactions (left), and of associativeand Penning ionization reactions (right), using the reaction scheme from tables B.1, B.2 and B.3, in the99.9% 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 → N+2 /N+

2 (B) and He+ → N+2 (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 reactionscheme 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 reactionscheme from tables B.1 and B.2 (left) and if 3-body reactions (R8), (R8b), (R14), (R14b) and (R22) areremoved (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 of100 (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.

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Figure B.5: Axial evolution of the e− densities for the cases of 10, 100, 1000 and 10 000 parts per millionof N2 in the He-N2 discharge at t = 2 µs and at a fixed radial position, from the model presented insection 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 permillion of N2 in the He-N2 discharge at t = 2 µs and at fixed radial position, from the model presentedin section I.3.1.

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