Modelling of atmospheric pressure argon plasmas:
Application to capacitive RF and surface microwave discharges
Mariana Atanasova (Pencheva)
Co-Promotor :
Prof. Dr E. Benova
A thesis for the degree of
Docteur en Science de l’ Ingénieur
2013
Aero-Thermo-Mechanic Department Physics Faculty
Université Libre de Bruxelles Sofia University
his world, after all our science and sciences, is still a miracle; wonderful, inscrutable, magical and more, to whosoever will think of it.”
Thomas Dekker
to my husband
T
I. General
1. Introduction 1
1.1. Brief history... 2
1.2. Basic principles of plasma... 3
1.3. Atmospheric plasmas. Classification... 6
1.4. Thesis overview... 8
2. Plasma regimes and role of the driving frequency 11 2.1. Introduction... 12
2.2. Heating mechanisms... 12
2.3. Operating modes and driving frequency effect... 13
2.4. Oscillatory aspects of plasma features... 17
2.4.1. The oscillation of the drift velocity... 18
2.4.2. The electron displacement... 19
2.4.3. The complex conductivity... 20
2.4.4. The electric power density... 20
2.4.5. The modulation depth of various quantities... 21
2.4.6. Applications... 22
2.5. Modelling requirements... 24
3. Atmospheric pressure argon kinetics 27 3.1. Collisional radiative model of argon at high pressure... 28
3.1.1. Energy level diagram of argon... 29
3.1.2. Electron excitation kinetics (EEK)... 32
3.1.3. Heavy particles kinetics... 40
3.2. Boltzmann equation... 47
3.2.1. Boltzmann equation for the electron energy distribution function... 48
3.2.2. Moments of the electron energy distribution function... 58
4. Mass flow description. RF CAP modelling 67
4.1. Introduction... 68
4.2. Model... 71
4.2.1. Configuration... 71
4.2.2. Chemistry... 74
4.2.3. Transport... 76
5. Plasma shower parametric study 79 5.1. Introduction... 80
5.2. Flowless non-perforated parallel plates: The plasma chemistry kinetics... 80
5.3. Flowless perforated parallel plates: Discharge characteristics in a non-flowing application... 84
5.4. Flow-flushed perforated parallel plates: The effect of the gas flow... 86
5.5. Driving frequency dependence of the plasma shower... 89
5.5.1. Macro effects... 89
5.5.2. Micro effects... 92
III. Surface microwave discharge
6. Electromagnetic wave description. SWD modelling 97 6.1. Introduction... 986.2. Model... 99
6.2.1. Physical situation... 99
6.2.2. General approach... 100
6.3. Electrodynamics of the wave propagation... 103
6.3.1. Symmetry... 108
6.3.2. Plasma surrounding media... 112
6.4. Plasma torch (cylindrical plasma–air configuration)... 113
6.4.1. Wave components... 113
6.4.2. Local dispersion relation... 115
6.4.3. Power balance of the wave... 116
6.5. Plasma in a dielectric tube (cylindrical plasma–dielectric–vacuum configuration)... 117
6.5.3. Energy balance equation for wave sustaining confined discharge... 120
6.6. Numerical procedure... 121
7. Cylindrical SW discharge parametric study 123 7.1. Introduction... 124
7.2. Wave propagation and field components... 126
7.2.1. From dispersion to propagation diagrams... 127
7.2.2. Criterion for the end of the plasma column... 132
7.2.3. The phase and group velocity... 134
7.2.4. Electromagnetic wave components... 136
7.3. Discharge characteristics as a function of the electron density... 141
7.3.1. General tendencies... 141
7.3.2. Dependence on the gas temperature... 144
7.4. Full description: Axial profiles of discharge features... 148
7.4.1. Dependence on the radius... 150
7.4.2. Dependence on the tube thickness... 155
7.4.3. Dependence on the electron–neutral collision frequency... 159
7.5. Comparison with experiments... 162
8. General conclusions 171
Acknowledgements 181
References 184
Summary
Objectives
Due to the ease of their use the variety of plasma sources at atmospheric pressure is already immense. They alter in terms of construction and operating conditions – working gas, power, gas flow as well as in terms of plasma characteristics – electron temperature and density, gas temperature.
This work is focused on modelling of atmospheric pressure high frequency (HF) discharges operated at relatively low power densities. Two types of devices are considered – the radio frequency capacitively coupled atmospheric pressure plasma jet and the microwave discharge sustained by surface electromagnetic waves. They are addresses as the plasma shower and the surface-wave discharge (SWD). Both of the considered devices operate in argon at atmospheric pressure (p = 1 bar). However, the difference in the frequency of the power coupling mechanism induces a big difference in plasma properties. This implies also that different modelling approaches have to be employed.
Modelling
In both model structures we follow the subdivision: configuration, transport and chemistry.
Configuration deals with the impact of the environment on the plasma and thus, among others, with the shape and sizes of the plasma, the boundary conditions and energy coupling modus. Transport describes the transport of species, momentum and energy in the plasma (and afterglow). Chemistry deals with the creation and destruction of plasma species; hence it has an important impact on the sources and the coefficients that drive and facilitate transport.
Shower plasma
A theoretical description of an RF capacitively coupled discharge with the form of a shower has been built. This device is quite complex. Apart from the sheath structure, typical for the parallel plate configuration, the performance of this plasma is also determined by the field augmentation and flow acceleration in the holes together with the flow recirculation in between the plates. The understanding of the concerted action of these ingredients was facilitated by two-dimensional spatial modelling. The model allowed us to investigate the role of the sheaths and space charge, the influence of the shower holes on the field distribution and the effect of the flow on the species profiles.
Surface-wave discharge
An intrinsic characteristic of SWDs is the interplay between exciting wave and resulting plasma. An adequate description requires a model accounting for these two aspects in a self-consistent manner. In this thesis we present such a model which consists of two interconnected parts. The first one is based on electrodynamics and provides a detailed description of the wave propagation. The second is based on chemical kinetics and enables studying plasmas at different discharge conditions. Combining them, a full description of SWD is achieved. By means of this model a study on the propagation characteristics of the surface wave maintaining the plasma and the field components as well as on the plasma characteristics – density, temperature, excited species distributions and etc., is performed.
The effect of the geometry and the high pressure conditions is also examined.
1
General Introduction
1.1. Brief history of the development of plasma research
A well known fact is that if we do not consider the dark matter (which nature is still unclear), over ninety percent of the matter in the Universe is plasma, the so-called "fourth state of matter." It is therefore not surprising that plasma has been and is currently a subject of numerous studies. Depending on the desired applications or fundamental knowledge sought for, these studies were developed in different disciplines.
The scientific approach of exploring the plasma state is established by Benjamin Franklin, who in the 18th century explored the natural atmospheric pressure discharge - the lightning. In the 19th century Michael Faraday studied glow discharges at low pressure.
The term "plasma" is used for the first time in physics to designate ionized gases as late as 1922 by the American scientist Irving Langmuir [1]. About that time, plasma physics as we know it today was born. Experimental studies were driven by the need of understanding the effects of ionosphere plasma on the propagation of radio frequency electromagnetic waves over long distances, as well as the development of gas-filled tubes used in the pre- semiconductor era of electronics for radio and television communication. In 1940 Hannes Alfvén developed the theory of hydro-magnetic waves, now known as Alfén waves and predicted their important role in the description of astrophysical plasma.
The 50s of the twentieth century marks the beginning of the development of charged particles accelerators, and therewith large-scale research on controlled thermonuclear fusion and the possibilities for its utilization as an energy source. Along with the work on fusion efforts are targeted towards extensive study on cosmic plasma. Nowadays the observation and theoretical modelling of plasmas in stars and other astrophysical objects, and the investigation of the interaction of the solar wind with the Earth magnetosphere have evolved as completely independent fields of research. Studies of engines aimed for adjusting the position of shuttles in space, which have began in the 60s and evolved into building of magneto-plasma-dynamic engines. These have the potential to make the interplanetary travelling possible.
After World War II plasma invades the field of material science as a powerful tool for surface deposition of layers, cutting and etching. These applications lead to the formation of a new area in plasma physics in the late 80s of the 20th century – called plasma processing. It is crucial for the production of tiny, complex integrated circuits used in modern electronic devices. Today this application is of particular economic interest.
From 1980 on the research on non-neutral plasmas emerges. The equations of their description resemble fully those used in the incompressible hydrodynamics. Therefore a well developed theoretical apparatus is available for modelling these plasmas. One of the important applications of non-neutral plasmas is the storage of large amounts of positrons.
The research on dusty plasmas began in the 90s. Dust particles immersed in plasma can acquire electric charge and act as additional charged particles. Since dust particles are massive compared to electrons and ions and can acquire various charges, their presence leads to a completely new behavior of the dusty plasma compared to "normal" plasma.
Furthermore, both non-neutral and dusty plasmas form eccentric, firmly connected states, which resemble solid state, i.e. quasi-crystalline structures. This phenomenon draws great interest.
In addition to the reviewed areas, there is an ongoing interest in the so called plasmas for industrial applications – plasma arc, plasma torches, laser plasma. For example, approximately 40 percent of the steel produced in the United States is processed for reuse in huge ovens where over 100 tons of steel waste can be melted in minutes by an electric arc.
Today, plasmas are used for sterilization, decontamination (cleaning) of water and air, wound and teeth treatment, etc. It allows development of new light sources; steps into our daily live in the form of plasma screens, projectors, and various multimedia and office equipment. Therewith, plasma continues to bring challenges both in fundamental understanding and in practical applications.
The field of research of this thesis is atmospheric pressure glow discharges induced by low power density. Created in open end tubes or directly in ambient air they feature low gas temperature but high reactivity. That is why this type of plasmas open up the possibility for numerous applications among which of most interest are the biological and biomedical ones. The focus is particularly on discharges created by means of high frequency electromagnetic field. Two types of discharges are studied – capacitively coupled plasma created by radio frequency field and plasma sustained by surface microwave.
1.2. Basic principles of plasma
What do we call plasma? Plasma is quasi-neutral gas consisting of charged and neutral particles which exhibit collective behaviour. As seen from this definition, not every ionized
gas is plasma. The term “quasi-neutral” means that regions with size larger than the so called Debye screening length are electrically neutral. Although consisting of numerous charged particles, plasma can be considered as electro-neutral, since the charges are mutually compensated. On the other hand there may be uncompensated charges within a sphere with the Debye screening radius, also called Debye sphere. The greater the density of charged particles, the smaller the Debye screening radius is.
The collective behavior typical for plasma is related to the presence of both inter- particle collisions, inherent for ionized gases, and of far-operating Coulomb forces. As a result, the motion of particles in the plasma is determined not only by local interactions, but also by interactions acting at greater ranges.
Third requirement an ionized gas must satisfy in order to be considered as plasma, is that its plasma frequency (i.e. frequency of oscillation of electrons as a result of perturbations of quasi-neutrality) is greater than the frequency of interaction between electrons and atoms. In this case the influence of electromagnetic forces on the behavior of the particles is substantial compared to the influence of hydrodynamic forces and the gas can maintain quasi-neutrality and exhibit collective behavior.
Shortly said, ionized gas can be called plasma when first, it is characterized either by sufficiently high density of charged particles or by high enough volume so that the radius of Debye screening sphere is much smaller than the physical dimensions of the system, second, the plasma constituents exhibit collective behavior and third, the frequency of electron oscillations is greater than the frequency of interaction between electrons and neutrals.
Depending on how it is created and maintained, plasma may be almost fully ionized (in which case electrons and ions have equal temperature, which is of the order of thousands of K) or it may contain just a small number of charged particles (electrons acquire high temperature but heavy particles are at about room temperature). These two types of plasma are called, respectively, thermal and non-thermal plasma, or more often, hot and cold plasma. It is this wide temperature (energy) range, in which plasma exists, that opens the door for such a variety of plasma technology applications.
The object of study in this work is non-thermal plasmas generated at atmospheric pressure. Generally, plasma is created by means of supplying energy to a gas. The energy absorbed by gas particles – atoms and molecules – changes their electron configuration creating excited particles, electrons and ions. The nature of this energy can be thermal,
electrical, gravitational or kinetic. In most laboratory plasmas the energy creating and sustaining the discharge is with electromagnetic nature and is directly coupled to the electron component. The reason is that electrons acquire a large mobility. In turn, electrons transmit the energy to heavy particles, such as atoms or molecules, by collisional interactions. These collisions can be divided into:
a) Elastic, which do not change the internal energy of the heavy particles and only increase or decrease their kinetic energy or translational motion;
b) Inelastic, which change the internal structure of the heavy particles, leading to birth of excited particles and electron–ions pairs.
Depending on the energy of the various plasma components and the rates of the ongoing interactions between them, plasma can be in thermodynamic equilibrium (equilibrium plasma) or to deviate from thermodynamic equilibrium (non-equilibrium plasma).
Thermodynamic equilibrium (TE) requires the rates of the forward and reverse elementary processes to be equal, i.e., any process of interaction is balanced by its opposite – excitation by deexcitation, ionization by recombination, and so on. This is the so-called principle of detailed balance. Moreover, the kinetic energy of particles is characterized by the Maxwell distribution function with corresponding kinetic temperature Tk, the density of excited particles is described by the Boltzmann distribution function with excitation temperature Texc, the density of ions – by the distribution function of Saha with corresponding ionization temperature Ti and the radiation density – by the distribution function of Planck with radiation temperature Trad. In thermodynamic equilibrium, these temperatures are equal and correspond to the thermodynamic temperature T [2].
Equilibrium distribution functions
• Maxwell: f (υ) ~ exp(–m υ 2/2 kTk)
• Boltzmann: f (Eqp) ~ exp(–Eqp/ kTexc)
• Saha: f (Eip) ~ (n/2) h3/(2 πm kTe)3/2 exp(–Eip / kTi)
• Planck: f (ν) ~ [–1 + exp(h v / kTrad)] –1
In thermodynamic equilibrium the description of plasma is relatively easy process, since for a given temperature, gas pressure and chemical composition all the characteristics that are of interest can be obtained from the distribution functions that are ruled by one and the same temperature. However laboratory plasmas deviate substantially from equilibrium.
The reason is that the small sizes and thus the large gradients present in plasma generate
large outward fluxes and there effluxes disturb the balances of forward and backward processes.
Especially plasmas at low power density, the object of this study, are far from equilibrium. But they might acquire sometimes a local or, more often, partial TE. In local TE the gradients of plasma properties – energy, density, thermal conductivity – are low enough to allow the particles in the plasma to reach steady state. Thus parameters vary in space and time, but vary so slowly that, for any point, one can assume thermodynamic equilibrium in some neighborhood around that point. This requires that the time for diffusion must be of the same or of greater magnitude than the time for which particles reach equilibrium. Partial TE exists when only some processes are in equilibrium. For instance, in plasmas with relatively low density electrons and ions are often with energies close to Maxwell distribution, but with different kinetic temperature, i.e. Ti ≈ Ta but Te ≠ Ti. That is because the exchange of energy in-between electrons is much more intensive than with the heavy particles, which is due to the difference in mass. In that case we can speak about partial equilibrium – separate for electrons and ions.
The question how real plasma is to be relevantly described requires an analysis on its particular state. Such an analysis should take into account what type of distribution functions can be applied to the different plasma components as well as the character of radiation and interactions between species.
1.3. Atmospheric plasmas. Classification
Because of the ease of their use, the variety of plasma sources at atmospheric pressure is already immense. They vary in terms of construction and operating conditions – working gas, power, gas flow rate as well as in terms of plasma characteristics – electron temperature and density, gas temperature, etc. If the energy coupling is with high power density, in general, the plasmas is with high electron densities. These high ne values induce frequent elastic and inelastic collisions. One of the consequences is that the gas temperature Tg can be high and approach the value of the electron temperature Te; thus Tg ≈ Te. Examples of these so called thermal plasmas are the plasma created in torches for cutting and welding and the atmospheric inductively coupled plasma (ICP), popular in the field of spectrochemical analysis.
At lower power densities the plasmas is away from equilibrium and for the kinetic temperatures the inequality >> is satisfied. Under certain conditions the gas
temperature may remain close to room temperature. These plasmas are denoted as cool or cold atmospheric pressure plasmas (CAPs). The plasma treated in this thesis can get properties close to those of CAPs. Thus this study can contribute to a better understanding of this booming field. The contemporary increase of interest in the CAPs is based on the fact that they hold out excellent prospects for technological applications. As James Schultz writes in his article Cold Plasma Ignites Hot Applications [3]
“Sterilization of food, medical equipment, and contaminated civilian and military gear is just one potential major application of so-called “cold” plasmas. These ambient-air- temperature ionized gases could also be used as a Star Trek-like protective shield around sensitive electronics-bearing devices, such as satellites; as cloaking technology for military aircraft, as a means of absorbing radar waves in order to remain hidden on enemy screens;
and as components of a new generation of miniature lasers and in advanced, low-energy- consumption fluorescent light tubes.”
One approach to classify atmospheric plasmas, both thermal and non-thermal, is according to their excitation frequency [4]. By this criterion, they can be divided into three categories: a) Direct current (DC) and low frequency discharges (1 kHz - 1 MHz), b) Radio frequency discharges (1 MHz - 1 GHz) and c) Microwave induced plasma sources (≥ 1 GHz). This leads to the more refined classification as given below:
• Direct current and low frequency discharges
o Continuous regime: the arc plasma torches [5,6]
o Pulsed regime:
Corona discharge [7]
Dielectric barrier discharge [8–10]
Corona-like discharges [11–13]
• RF discharges
o High power discharges
Inductively coupled plasmas (ICP) [14,15]
RF pulsed discharge – IST system [16]
o Low power discharges
Capacitively coupled plasma jet (APPJ) [17–19]
Cold plasma torch [20,21]
Hollow cathode discharge [22,23]
• Microwave induced plasmas (MIPs)
o Resonant structures [24–26]
o Coaxial structures (TIA from French Torche a Injection Axiale) [27–29]
o Microwave torch discharge (MTD) [30]
o Microwave plasma jet (MPJ) [31]
o Microwave plasma torch (MPT) [32–34]
o Surface wave sustained discharge (SWD) [35–41]
o Microwave plasma source (MPS) [42]
In this thesis we will focus on two of these devices – the RF capacitively coupled atmospheric pressure plasma jet and the microwave discharge sustained by surface electromagnetic waves. Further on, they are going to be address as RF CAP or plasma shower and SWD. When the validity of the considerations brought up in the thesis applies for both discharges we will use the term high frequency (HF) discharges. Both of the considered HF devices operate in argon at atmospheric pressure (p = 1 bar). However, the difference in the frequency of the power supply determines different properties of the plasma and thereafter potential application.
Argon plasma was considered because, as the experiments show, it has better energy transfer efficiency compared to helium plasma under the same working conditions. It also shows better performance from a practical point of view in terms of gas consumption and treating time [43,44]. And finally it is less expensive, easy to produce and conserve.
The boost of interest in CAPs is mainly based on the flexibility of these sources for generation of strongly non-equilibrium plasma conditions in an economic and reliable way in open air. This is one of the main reasons why CAPs continuously find new applications.
The RF CAP device is a discharge that features low (close to room) gas temperature, that is versatile in selecting the gas chemistry, and adjustable in power input and gas flow rate. These advantages result in applications such as surface treatment of polymers, surface modification, film coating and etching, developing of nanotechnologies, environmental applications, sterilization and wound healing [45–50].
The SWD CAP offers a great variety of scientific and technological applications - in mass spectrometry for the analytical determination of substances, in surface functionalization [51–54], lasers [55–57], lighting [58–61], etc. Moreover, they have specific advantages over the other microwave plasmas. For example, an important advantage of SWD used as excitation sources in atomic emission spectroscopy is the efficient excitation of non-metals, which are not readily accessible for detection [62].
Another advantage in comparison with other electrodeless discharges is their higher electron plasma density at a given input power. In addition, they are effective sources of chemically active particles – ions and excited atoms and molecules [63,64].
However, high pressure devices tend to undergo contractions and instabilities.
Therefore, for the purpose of stabilization, a reduction of discharge dimensions is often necessary. Depending on the application, the width of the discharge can range from less than 1 mm to about tens of mm. The latter is the range in which the discharges under study in this thesis operate.
1.4. Thesis overview
This thesis is composed of three main parts. Its structure and the content are as follows.
The first part is devoted to general aspects of atmospheric plasmas. Chapter 2 gives an introduction into plasma generation and the role of the driving frequency on the discharge maintenance and properties. Attention is also given also on how this effects the modelling of the two types of plasma, which are object of this thesis. The frequency, which is appropriate in a particular case, is determined by the application: in general one needs to promote either charged particles (electrons and ions), or neutrals (atoms, molecules or radicals). The determination of the optimal species concentration requires a thorough knowledge of the part played by those species in the application, and further, of the cross- sections of the processes leading to production and destruction of these active species as well as of their transport parameters. All this is discussed in chapter 3, presenting the collisional radiative model built for atmospheric pressure argon.
The second part is devoted to the RF CAP device, the so called plasma shower. Chapter 4 presents the model in details. It provides a description of the device’s configuration and the modelling approach. The explanation of the chemistry, which involves the species treated in the model, the elementary processes considered, as well as all the particles’
transport and rate coefficients, is given there. The transport generated by the drift, the diffusion and the convection of species is treated as well. Chapter 5 presents the modelling results. First the numerical model is validated by means of simulations of the widely studied parallel plates’ configuration (section 5.2); the results are compared with the available data from experiments and other models. A discussion on the role of the chemistry kinetics is also conveyed there. Then the shower device is modeled and the obtained distributions of the species and the plasma potential for the case of a non-flowing
application are presented in section 5.3. The role of the flow on the discharge properties is examined and discussed in section 5.4. The next section 5.5 focuses onto the role of the frequency of the driving field on the plasma features of the parallel plate configuration, particularly in relation with the evolution of the densities of various active species.
The last part of the thesis considers the SWD. Chapter 6 gives a detailed description of the electromagnetic surface waves. It presents the main electrodynamic equations as well as the boundary conditions used for obtaining the wave propagation characteristics. It also discusses how to build a self-consistent axial model of argon stationary state discharge at atmospheric pressure produced by means of a surface-wave. Chapter 7 gives the results obtained applying the model. Two set-ups are considered, plasma–vacuum (air) and plasma–dielectric–vacuum as explained in the introductory section. First the propagation of azimuthally symmetric wave mode is analyzed. The analysis includes dispersion and attenuation diagrams applicable for uniform medium as well as propagation and damping diagrams applicable in the non-uniform case (section 7.2). The surface wave field components are obtained and shown as well. Then the discharge characteristics are derived as function of the electron density (section 7.3). The effect of gas temperature, being a parameter which is not self-consistently calculated, is studied. In section 7.4 by means of the full model the axial dependences of the wave and plasma features are obtained and presented. The effect of discharge radius and tube wall thickness as well as the role of the collisions between electrons and neutrals on both, the characteristics of the wave maintaining the plasma and the properties of the plasma itself, are investigated and discussed. Finally, in section 7.5 the results are compared with the available experimental data and a discussion on the validity of the model is given. The thesis ends with chapter 8, in which the main conclusions are summarized.
2
Plasma Regimes and Role of the
Driving Frequency
2.1. Introduction
From engineering point of view there are many different ways a plasma source to be constructed. But in most cases there are some general conditions that have to be fulfilled by all of them. As the temperature of the plasma is larger than that of the environment we can expect that plasma will continuously lose energy to its surroundings. Therefore there must be energy input to produce and sustain the plasma. In stars the driving energy is gravitational or nuclear in nature. For plasmas created during the re-entry of space vehicles it is the kinetic energy of the space craft and in the flames the nature is chemical. However, in most of mankind created plasmas the energy source is electromagnetic in nature. This implies that the conditions for the plasma maintenance are essentially determined by the charged particles and the energy exchange mechanisms. The method of sustaining the plasma from an external energy source determines different heating mechanisms, which in turn define different regimes of discharge functioning and therewith plasma characteristics.
In the category of EM energy driven plasma we can make a further sub-classification that mainly depends on the frequency of the EM field. The present chapter is intended to provide some background on the variety of heating mechanisms in HF discharges, the various modes in which plasma operates, and the effect of the driving frequency.
2.2. Heating mechanisms
In this thesis are considered two ways of transferring energy from the EM field to plasma discharges. The resulting electron heating mechanisms are Ohmic heating and stochastic heating [65].
Ohmic heating is present in all EM driven discharges. The basic mechanism is the transfer of systematic energy gained by the electrons from field acceleration to random, thus thermal, energy through local collisional processes. It is particularly important at high pressure at which the collision frequency is high. At atmospheric pressure it is the dominant heating mechanism.
Stochastic heating is a mechanism typical for capacitive discharges. Electrons impinging on the oscillating sheath-edge suffer a velocity-change upon back-reflection into the plasma bulk. As the sheath moves into the bulk, the reflected electrons gain energy;
when the sheath moves away, the electrons lose energy. However averaging over an oscillation period, there is a net energy gain. The reason is that the number of electrons
encountering the sheath-edge expanding within the bulk is larger than when the sheath edge shrinks back.
Secondary emission, especially in DC and capacitively couple RF discharges, can play a crucial role in plasma production and can also contribute substantially to electron heating.
It is fundamental to the operation of DC glow discharges.
Wave–particle interactions is a fundamental method of transferring energy from fields to electrons and are an important mechanisms of electron heating in high density discharges such as SWD sources. At high pressure this heating is collisional, i.e. Ohmic heating. If collisions are negligible, the electrons follow the oscillation of the field with a phase delay of π/2 and on average there is no dissipation of the wave energy into the plasma. However, when collisions between electrons and plasma particles occur, the collisions cause a phase change in the electron motion, resulting in a net energy transfer from the electric field to the plasma. At low pressure the energy transfer can be also collisionless, i.e. a resonant one.
2.3. Operating modes and driving frequency effect
Let us consider a classical glow discharge sustained between two parallel electrodes placed in a gas-filled vessel by a DC power supply (figure 2.1) [66].
Figure 2.1. Classical glow discharge created between two parallel metal electrodes by means of DC power supply.
When a sufficiently high potential difference is applied between the two electrodes the gas will break down into positive ions and electrons, giving rise to a gas discharge. In a gas there are always some few free electrons available created by cosmic radiation. However, without applying a potential difference, the electrons are not able to create and sustain a discharge. When a potential difference is applied, the electrons are accelerated by the electric field between the cathode and anode and collide with the gas atoms, which may
E
C A
− +
result in a gas breakdown. The most important interactions are the inelastic collisions, leading to excitation and ionization and thus plasma creation. The ionization collisions generate new electron-ion pairs. On their way toward the anode the newborn electrons may gain enough energy to perform ionization, and in turn, collide with the gas neutrals giving birth to new electrons and ions. This chain process of volume ionization is called electron avalanche. It is a key process in low pressure discharges, but it is rather negligible at atmospheric pressure. The reason is that due to the high collisionality present in high pressure the electrons can not gain enough energy in a collision free path. In contrast, the group of electrons is heated and exchanging their energy by means of Coulomb collisions some get more energy than others. The high energetic ones, those in the so called tail of the distribution, can perform ionization processes or chains of excitation processes that lead to ionization. This shows that electrons perform essential mechanisms that are needed to generate and sustain plasma. However, especially in DC or low frequency discharges we can not neglect the special action of the ions; they are needed to get the electrons into the discharge. The ions are accelerated by the electric field toward the cathode, where they can release new electrons by ion-induced secondary electron emission. These highly-energetic electrons give rise to new ionization collisions, creating new ions and electrons. This mechanism of sustaining a plasma, called secondary electron emission, is also known as the γ-mechanism [67], named after the ion-induced secondary electron emission coefficient γ.
As said, in DC discharges the γ-mechanism is indispensable. The electrons pumped by the power supply go through the wires (made for instance by Cu) towards the cathode, via the plasma to the anode where they render back to the wire. At the cathode electrons have to make a „material jump” – a change – from Cu into plasma. This can only be done by ion assistance; ions have to release electrons from the cathode. And they do that via bombardment, i.e. the γ-mechanism. Of course other mechanisms, like photo- or thermionic secondary electron emission are also possible. But these come into play for plasmas/discharges with high energy density.
This γ-mechanism takes place in the cathode fall (or sheath), a capacitor-like layer of the cathode over which a considerable voltage drop takes place. Due to the difference in the velocities of electrons and ions the potential difference applied between the two electrodes is generally not equally distributed. Thus by charge movement the self- organization of the plasma results in a large voltage drop over the sheath, creating a high
Figure 2.2. Plasma potential in the sheaths and in the bulk of the plasma. It is always positive with respect to the adjacent walls in order to reduce the electron loss rate toward the walls.
If instead of a DC an RF voltage is applied to the electrodes, high voltage capacitive sheaths are created between the two electrodes and the bulk of the plasma. The RF currents, flowing across the sheaths and through the bulk of the plasma, lead to ohmic and stochastic heating in the sheaths and ohmic heating in the bulk.
In the capacitive discharge, i.e. for higher frequency of the exciting field, the „material jump” is no longer needed; the electrons do not go around the circuit but make local oscillations. During the oscillations collisions with atoms and molecules resulting in ionization take place: the α-mechanism. This ionization during oscillations keeps the creation of electrons on-going.
Gas ionization in the α-mode is volumetric occurring throughout the plasma, in contrast to the γ-mode, where it is dominated by events localized near the boundary between the sheath and the plasma bulk. In the α-mode the electron energy acquired in the sheath is small, which suggests that many electrons in the sheath region have a kinetic energy below the ionization threshold and their acceleration needs to be continued in the plasma bulk in order for their kinetic energy to reach high enough for ionization value. This is the reason why electron production is volumetric. Both the volumetric electron production and the small electron mean energy in the sheath are typical characteristics of the α-mode [68,69].
It is worth mentioning as well that in α-mode there is a gradual voltage dependence of the electron density. This results in relatively easy control of plasma stability and this is why most reported experiments of the RF CAPs are operated in this mode [19,20,70,71].
The modelling of these type discharges in the current thesis is also restricted to that case.
It is known that in α-mode secondary electron emission does not significantly influence the electron density and other characteristics. The primary electrons accelerated by the oscillating field of the applied voltage dominate the ionization. The effects of secondary
C A
− Vp +
electrons are insignificant when RF APGD operates with low current densities [70], i.e. α- mode.
However, if the discharge is forced to generate a larger current it might again ask assistance from the γ-mechanism. This is the α - γ transition and depends on the current I or more precisely on the current density j.
In the high current case, i.e. the γ-mode again, the maximum electron mean energy (in the sheath) is found to be much larger than in the α-mode. Therefore, compared to the α- mode case, significantly more electrons can reach the ionization energy within the sheath, thus resulting to a situation in which most ionization events are confined to the sheath-bulk boundary region. The more frequent ionization events in the sheath suggest a more rapid energy loss among more energetic electrons, leading to a smaller electron mean energy in the plasma bulk compared to that in the α-mode case.
The secondary electron emission plays a crucial role here. With a large secondary emission coefficient, more electrons are available to be accelerated to the ionization energy within the sheath, thus resulting in a large space-charge field to be established at a relatively low RF voltage [67], and thereafter the high mean electron energy achieved there follows.
The α – γ mode transition is a result of a sheath breakdown [72]. It occurs when the voltage across an electrode sheath exceeds the breakdown voltage of a gas gap of the same size as the sheath thickness.
Going to even higher excitation frequencies the plasma operation changes substantially and EM wave aspects become important. In RF discharges the wave length of the field is much higher than any dimension of the discharge. Thus it is not necessary to account for the wave propagation. However, for frequencies above 100 MHz, i.e. MW discharges, the wave propagation becomes important. In the case of MW discharge sustained by travelling electromagnetic wave, considered in this thesis, the wave propagation is a key feature.
Plasma does not permit bulk propagation of an electromagnetic wave when the excitation frequency is less then the plasma frequency given by the electron density of the plasma. In atmospheric pressure plasmas it is usually the case as the plasma frequency is of the order 1010 Hz. However, at these frequencies plasma still can support a so called surface-wave along the interface of the plasma with vacuum or a dielectric material (for example, a surrounding wall) [74,75]. Given the wave frequency, propagation of a surface
from the condition ωp = ω. For electron densities higher than the critical density, a surface- wave can exist with its phase-front normal to the interface and it has an electric field component that is parallel to the propagation direction. The field amplitude within the plasma reduces to 1/e over a distance δ, called skin depth.
A surface-wave discharge can be started by applying a field in the direction parallel to the interface that contains enough energy to ionize the gas further by accelerating the already available free electrons. At a given moment, the electron density exceeds the critical density and a plasma wave with frequency ωp > ω is excited within the plasma, while the electromagnetic wave is guided as a surface-wave along the interface to some point away from the location where the microwave field is applied. In this manner the surface-wave discharge gradually extends along the dielectric wall until there is not enough energy left in the wave to ionize the gas at that position to a level beyond the critical density. Because of this self guiding principle of the surface-wave discharge, it guarantees a minimum electron density, based on the field frequency chosen: The higher the field frequency the higher the critical density is for the surface wave discharge to exist [76].
Apart from the electron density ne and the field frequency ω, surface-wave discharges are characterized by the electron temperature Te and the effective collision frequency ν.
There are various methods to determine their values experimentally, such as measurement of the reflection coefficient of microwaves [77] and spectroscopic methods [78]. Another approach to determine these parameters is to construct a detailed discharge model that takes into account electrodynamics of the wave propagation and particle kinetics [79].
The total length of the surface-wave discharge decreases with increasing field frequency at constant input power, because of the increasing electron density required for propagation. The discharge length decreases also with increasing collision frequency because of the increasing dissipation by collisions but increases with the applied power.
Finally, the discharge length depends on the nature of the gas, mainly because of differences in ionization energy.
2.4. Oscillatory aspects of plasma features
In the previous section a qualitative description of the discharges as function of the driving frequency fd was given. This section aims to give a more quantitative description of the various oscillatory aspects of plasmas driven by a high frequency. We keep in mind that
the plasmas in this study are either driven by RF, with a frequency value fd = 13.56 MHz or by microwave, for which fd = 2.45 GHz.
The driving frequency, in first instance, leads to oscillatory behavior of charged particles: electrons and ions. This is manifested in a periodic displacement r(t) of electrons and ions but also in periodic velocities v(t). On a bigger scale this leads to oscillatory behavior of the current density j(t) and thus the density of the electric power dissipation Q(t). In this section it is investigated how these quantities, r(t), v(t), j(t) and Q(t), depend on the driving frequency and how the oscillatory behavior of the power density Q(t) influences other, more fluid like, plasma aspects.
2.4.1 The oscillation of the drift velocity
The equation of motion (EOM) of a mean electron subjected to a periodic E-field reads
me ∂t v(t) = − eE(t) − meνv(t)
where ν is the frequency for momentum transfer from the electrons {e} to the heavy particles {h}, me – the mass of an electron, whereas v(t) and E(t) are the temporal vareable velocity and electric field. Note that v(t) is the systematic or drift velocity and should not be confused with the thermal velocity vth. For the considered plasmas in most cases
|v| << vth. Thus the above equation gives the relation between the drift velocity of an electron subjected to the electric force and a “friction” force due to e-h collisions. In the analysis E(t) is taken pure harmonic, i.e. E(t) = Re{Ec exp(iωt)} with Ec – the complex amplitude of the E-field. Substituting E(t) = Ec exp(iωt) together with v = vc exp(iωt) into equation 2.1 gives
vc = − eEc [meν (1+ iω/ν)]−1
It shows how the complex amplitude vc depends via ω /ν on Ec. If ω /ν is small, i.e. if friction dominates vc ≈ − eEc[meν]−1 follows the field force –eE with an amplitude that does not depend ω. However if ω /ν is large vc ≈ − eEc[meiω]−1 lags behind while the amplitude decreases with increasing ω.
(2.1)
(2.2)
Thus we can conclude that a relatively low frequency (or high friction) results in good following at constant amplitude. A relatively high frequency results in bad following; there is a phase lag of 900 while the amplitude decreases with ω.
Expression 2.2 can be employed to obtain information of the current density j(t) and the power density Q(t) but it can also be used to get insight in the amplitude of the motion of ions and electrons. Let us start with the latter.
2.4.2 The electron displacement
The relation between the complex amplitudes of the displacement and the velocity, being vc = iωrc, can be used to find the amplitude of the displacement of an electron
rc = − eEc [meiω(ν + iω)] −1 or rc = µeEc[iω(1 + iω/ν)] −1,
where µe = − e/(meν) is the electron mobility (note the minus sign). For small ω (or large friction) limit ω << ν we the amplitudes is given by the relation r = µeE(ω)−1. The larger the friction (thus ν value) the smaller µe and thus |rc| is; but an increase of ω gives an extra reduction of the amplitude of the displacement.
For atmospheric pressure plasma the collision frequency is high ν ~ 3 × 1010 Hz.
Compared to the RF driving frequency fd = 13.56 MHz and thus ω = 8.5 × 107 Hz this is about ν /ω ~ 350 times larger. In the microwave case fd = 2.45 × 109 Hz and thus ωµw = 1.54 × 1010 Hz, thus again ν > ω but the now the ratio ν /ω ~ 2.
A few interesting features can be noted here. The above equation shows that the amplitude decreases with increasing frequency. Therefore in the case of the microwave the electrons stay at their location. For the RF case it is mainly the collision frequency that limits the motion of the electrons. But it is possible electron drift losses to be present if rrms > L. Thus, if the amplitude is larger than the length of the discharge electrons are lost via the external circuit. This demands for the γ-mechanism.
The above was focused on the electron behavior, however the relations can be used for ions by replacing µe with µi; but since µi << µe the corresponding rrms is much smaller; only for low frequencies (i.e. ω values) the ion will respond oscillatory to the driving frequency.
In many cases the ions can be considered as standing still while the electrons perform oscillations.
(2.3)
The dependence of the displacement rrms on the periodic voltage with amplitude V0
applied over the discharge with length l equals rrms = µe (V0 /l) ω−1. This parameter is in particular useful to study the starting up phenomena of discharges. But it can also be applied to the width of and voltage drop over the sheath of “settled” plasma.
2.4.3 The complex conductivity
The above has shown that the systematic motion of ions can be neglected so the current is carried by electrons, meaning that the current density reads: j = −e ne vc. Together with equation 2.2 this leads to
jc = e2 ne Ec[me(ν + iω)]−1 or jc = σEc
where the conductivity,
σ = e2 ne [me(ν + iω)]−1
being a complex quantity can be written as σ = σre + i σim with
σre = e2 ne ν / [me (ν2 + ω2)] and σim = − e2 ne ω / [me (ν2 + ω2)]
For the RF case we have σim << σre while the latter reduces to σre = e2 ne / [me ν]. In the microwave case both σim and σre are important and the ω2 term in the denominator has to be retained since ωµw is on the same order as ν.
2.4.4 The electric power density
The electric power density Q can be obtained by the multiplication
Q(t) = E(t) j(t) = Re (Ec exp( iωt)) Re (jc exp(iωt))
After some algebra this leads to ε(t) = 1/2 Re(σ) E2 which, based on equation 2.6, gives Q(t) = 1/2 {e2 ne ν / [me (ν2 + ω2)]} (1 + cos 2ωt) E2.
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
So the temporal behaviour of Q(t) is determined by the factor 1/2(1 + cos 2ωt) that oscillates with double frequency around the value 1/2. The modulation depth is thus 100%.
This modulation will effect other plasma quantities like the electron temperature Te and via this possibly the radiation emitted by excited levels. This will be investigated below.
2.4.5 The modulation depth of various quantities
The study of the influence of the modulation depth on other plasma quantities is guided by the differential equation
m ∂tq + ξq = F(t),
It is inspired by the EOM (cf. equation 2.1) but is more general in the sense that F(t) is now an arbitrary driving quantity for instance the electric field or power density and q – a quantity that is following, such as the velocity (like in equation 2.1) or the temperature.
The m and ξ are (generalized) mass and friction coefficient. For the driving quantity we suppose a form
F(t) = F + F*(t)
with F constant in time whereas F*(t) is the harmonic part ruled by Fc exp(iωt). The modulation depth of the driving quantity is now Fc/F. We can expect that q will have the form q = q + q*(t), where q is constant while q*(t) is described by qc exp(iωt). Substitution into equation 2.9 gives
m iωqc exp(iωt) + ξ q + ξ qc exp(iωt) = F + Fc exp(iωt).
which gives two independent equations
qc = Fc [ξ(1 + m iω/ξ)] –1 q = F/f.
(2.9)
(2.10)
(2.11)
(2.12) (2.13)
The first expression is in essence the same as equation 2.2 and shows the importance of the dimensionless quantity mω/ξ, telling that at relatively low ω values a quantity (subjected to a relative large friction) will follow the driving quantity with a ω-independent amplitude namely qc = Fc/ξ and that at relative high frequency (with respect to friction), we get a lag- behind 900 with an amplitude qc = Fcξ/miω that decreases with ω. The modulation depth is obtained dividing equation (2.12) by (2.13)
qc/q = (Fc/F)[(1 + miω/ξ)]–1.
has the same dependence. And we can conclude that a low frequency and/or high friction results in good following at constant modulation depth; high frequency and/or low friction results in bad following with decreasing modulation depth while the q lags 900 behind F.
2.4.6 Applications
Dependence of electron energy distribution function on frequency
The electron velocity distribution function f(v) describes the probability density of finding electrons in given interval of velocities v, v + dv. In the presence of harmonic electric field, given by E exp(iωt), it depends on electron velocity and the angle χ between the velocity direction and that of the electric field. Using the first order spherical harmonic expansion in velocity space and a first order Fourier expansion in time, one obtains
f(v) = f0(v) + v/v . f1(v) exp(iωt) = f0(v) + f1(v) cos χ exp(iωt).
The isotropic component f0(v) is independent of time. It determines the average value of the electron energy, and serves to calculate the average value of any quantity depending only on the absolute value of the electron velocity and not on its direction, e.g. kinetic energy. In chapter 3 we shall show that it is directly related to the electron energy distribution function (EEDF). The component f1(v), called the anisotropic term, determines the average value of electron velocity under the influence of electric field E (which is treated as a complex quantity). In a harmonic field the two components of the distribution function are related by
(2.14)
(2.15)
f1(v) = eE/me ∂f0(v)/∂v [νc(v) + iω]–1
Equation (2.16) is valid only under certain conditions. For instance, the isotropic component of f(v) can be modulated in time in some cases. Then, in order to account for this modulation, the Fourier expansion must be extended beyond f0(v) to higher orders.
Here we shall examine under what conditions these higher orders can be neglected.
The relaxation of the electron energy ε = mev2/2 through electron–neutral collisions is described by the characteristic frequency for energy transfer νε
νε(ε)= (2me/M)ν(ε) + Σνk(ε)
where M is the atomic or molecular mass and νk(ε) is the electron–neutral collision frequency for atomic or molecular excitation to energy level k. It can be shown that the time evolution of the isotropic component is determined by the relative values of ω and νε(ε). When ω << νε there are so many collisions during one period of the field that the energy accumulated by the electron between two successive collisions is closely related to the instantaneous value of the HF field intensity. Then the isotropic component is modulated over its entire energy range at twice the frequency ω, since it depends on the square of the electric field. Increasing ω the amplitude of this modulation decrease and it becomes negligible when νε < ω. For atomic gases it is the bulk of the distribution function that first stops to oscillate when ω increases because νε(ε) is smaller there than in the tail (see equation 2.17), where inelastic collisions take place.
And so EEDF is nearly stationary when νε < ω. To get an impression on the quantities we concentrate on the elastic collisions for we found for the frequency for momentum transfer equals typically ν = 3×1010 Hz for atmospheric conditions. Since 2me/M = 2.7×10-5 we find for νε = 2(me/M) ν = 8.1×105 Hz. In the RF case we see that with ωRF = 8.5×107 Hz the condition νε < ω is fulfilled. For the microwave the condition νε < ω is certainly fulfilled.
Harmonic E-field: the wave length λλλλ
When considering a harmonic E-field an important characteristic is the wavelength of the field. At low frequencies the wavelength effects, i.e. the role of λ, can be neglected.
Everything takes place at the first local “E wave crest”. For instance at f = 50 Hz the λ0 = (2.17) (2.16)
c / f = 3×108 / 50 = 6×106 m for waves in vacuum. This is much larger that the dimension of any laboratory plasma.
At higher frequencies λ comes into play, meaning that large E variation is caused by wave over plasma size. For example substantial part of the thesis is based on f = 2.45 GHz, for which the corresponding wave length we find is λ0 = 3×108 / 2.45×109 = 12 cm.
Also wave damping has to be considered and this strongly depends on pressure/chemistry.
Below we will confine ourselves mainly to the atmospheric conditions and Ar.
Propagation of wave in plasmas will change the wavelength; in most cases it is no longer given by the vacuum relation λ0 = c / f. In general λ is a function of f. This is known as the dispersion relation, a term that stems from the study of the change in wave propagation (say λ) as a function of driving frequency. This is usually done for a uniform medium for instance the ionosphere. Thus the waves in the constant medium are studied by varying the frequencies. In laboratory plasmas strong uniformities are present and it becomes interesting to see how the wave changes upon traveling through the changing medium. So instead of studying the constant medium with variable frequencies we study waves of a constant frequency in a variable medium. In the later case the λ(f) relation is usually denoted by the phase diagram.
The relation between the temporal and spatial frequency will be dealt in chapter 6 using propagation diagrams. Another spatial frequency is the (reciprocal) damping length.
2.5. Modelling requirements
Modelling of HF discharges requires first of all a description of the electromagnetic field in the plasma. However, to determine self-consistently the field in the plasma, is a quite complex task, even for the simplest situations. It means solving of Maxwell equations, which require knowledge on the currents and volume charges in the discharge determined by the plasma properties. But in order to determine the plasma characteristics and their profiles it is necessary to solve the kinetic equation for electrons and ions, for which we need to know the electromagnetic field distribution in the plasma.
The discharge field generally consists of quasi-stationary and high frequency components. In the absence of an external magnetic field the quasi-stationary electric component is a field that can be described by a scalar potential ϕ. This field tends to maintain the charge neutrality and is low in the bulk of the plasma and quite high within
magnetic field on the motion of plasma species is small. Thus it is mainly the electric component that is responsible for the electron heating and hence for discharge sustaining.
If the phenomena related to wave propagation are negligible, the magnetic field created by the displacement current can be ignored and the Maxwell equations can be confined to Poisson’s equation for the potential field ϕ [80]. This is the case of the plasma shower discharge considered in this thesis. On the other hand the modelling of the surface wave discharge (SWD) requires a solution of the full set of Maxwell equations.
Let us outline the main features of SWD and RF CAPs important for their computer modelling.
Capacitively coupled plasmas (CCP), like the plasma shower, work on electrostatic principles; the functioning of a surface wave discharge, on the other hand, is based on electrodynamics, i.e. the propagation of EM waves. Thus for a CCP Poisson’s law is essential, while the description of the electromagnetic behaviour of SWD is based on the combination of Faraday’s law and the Amper-Maxwell equation.
The shower is an electrode-driven discharge; the surface wave plasma on the other hand is electrodeless. In the RF discharges the sheath next to the electrodes plays an important role in accelerating ions and creating secondary electrons upon collisions with the cathode.
This implies that space charges play an important role in the shower plasma, while the surface wave plasma can be treated as being quasi neutral. This difference is among others related to the (mean) electron density. The shower plasma is characterized by low electron densities (1013–1019 m–3), while the surface-wave plasma acquires high plasma densities (1018–1021 m–3). The existence of electrodes in the plasma shower implies that surface processes such as secondary electron emission and/or sputtering have to be taken into account. This is not needed for the surface wave plasma.
In a CCP the electron density and the sheath voltage follow the oscillations of the externally applied electric field and change both in time and space. While in a SWD the wave frequency ω is comparable to the electron plasma frequency ωp in the bulk plasma;
close to the plasma boundary we even have ωp < ω. And since the time scale for the sheath formation is ωp–1, it follows that the sheath in a SWD remains largely unchanged during one time period of the oscillating EM field. For modelling this means that in a SWD one can consider the electron density and plasma potential as being constant in time [81] so that a stationary model can be used. Moreover we can work in a SWD with space averaged plasma quantities. On the other hand the CCP requires a time-dependent model and for the
