Dust particle(s) (as) diagnostics in plasmas

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Dust particle(s) (as) diagnostics in plasmas

Citation for published version (APA):

Beckers, J. (2011). Dust particle(s) (as) diagnostics in plasmas. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR719541

DOI:

10.6100/IR719541

Document status and date: Published: 01/01/2011 Document Version:

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Dust Particle(s) (as) Diagnostics in Plasmas

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Typeset in LATEX2ε using the Winedt editor.

Cover design by Rosalie Kibbeling. Original cover painting by Bas Beckers:

"Free interpretation of the several aspects elaborated on in this thesis; particle inter-action with a laser beam observed by the eye of the experimentalist under hyper- and microgravity conditions (2011)."

Printed by the Eindhoven University of Technology PrintService, Eindhoven.

This research was financially supported by the European Space Agency (ESA) and the Netherlands Space Office (NSO).

A catalogue record is available from the Eindhoven University of Technology Library Beckers, Job

Dust Particle(s) (as) Diagnostics in Plasmas / door Job Beckers. –Eindhoven : Technische Universiteit Eindhoven, 2011. –Proefschrift.

ISBN 978-90-386-2940-7 NUR 926

Trefwoorden : plasmafysica / stoffige plasma’s / koolwaterstof / micro-zwaartekracht / hyper-zwaartekracht / elektrische veldsterktes.

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Dust Particle(s) (as) Diagnostics in Plasmas

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof.dr.ir. C.J. van Duijn, voor een

commissie aangewezen door het College voor Promoties in het openbaar te verdedigen op dinsdag 20 december 2011 om 16.00 uur

door

Job Beckers

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1

Introduction

Plasma is everywhere: most of the visible matter in the universe (99%) is in plasma phase. Examples are interstellar clouds, stellar coronas and stars like our sun. Also on earth, there is a large variety of plasma appearances, both natural and human-made. Natural plasmas occur in the form of lightning and aurora. Man-made plasmas have an enormous range of applications in the solar cell and semiconductor industry, as light sources, in nuclear fusion devices etc. In this chapter we briefly introduce the concept plasma (section 1.1). Then, in the sections 1.2 and 1.3 we give a brief overview of the two research areas relevant for this thesis; dusty plasmas and the exploration of sheath phenomena using microparticles. Finally, in section 1.5 we define the objectives of this research and give an overview of the structure of this thesis.

1.1 Plasma as the fourth state of matter

Plasma is often referred to as the fourth state of matter, next to the well-known other three states: solid, liquid and gas. At low temperatures, matter appears as solid. The small components (atoms and molecules), from which material is made up, attract each other strongly and are packed together in certain structures. When (thermal) energy is supplied to the material, the mean kinetic energy of the components (the temperature) increases and as a result the bonds between them get weaker. When the increase in temperature is sufficient, the solid material melts and becomes liquid. In this state the components still feel

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attraction to one another. However, this attraction is not so strong that they are forced into the structure manifesting the solid state. When the temperature is increased to even higher values, their kinetic energy becomes so high that the components lose their mutual attraction. At this moment, the components can freely move through space and do not interact with each other1_{, i.e. the material}

is now in the gas phase. In order to explain plasma as the fourth state of matter, we review the Atomic Model of Bohr in which he proposed a quantized shell model describing that the electrons – orbiting the positive nucleus of an atom – are restricted to certain fixed orbits with fixed distances to that nucleus. When the material is in its gas phase and the internal energy of the atom is increased even more, one of the electrons in the outer shell might become released from its orbit and is free to leave the atom. This free electron leaves behind a positively charged atom, named an ion. The process of an atom losing an electron and being turned into an ion, is referred to as ionization. When a gas contains so much internal energy that (a part of) the atoms are (is) ionized, it is called a plasma. It was not until 1879 that Sir Williams Crookes identified the plasma state in a Crookes tube and called it ’radiant matter’. However, the term plasma was introduced by Irving Langmuir in 1928, because this state reminded him of how blood plasma carried its components [1].

1.2 Dusty plasmas

A dusty plasma is a plasma that not only consists of the regular neutrals, elec-trons and ions, but also contains nanometer to micrometer sized solid particles. These particles can be inserted into the plasma, but can also be created from chemical reactions between components in a chemically reactive plasma itself. Formation and growth processes of nanometer to micrometer sized dust parti-cles are increasingly frequently investigated for their broad range of applications. For example, dust particles created in silane discharges have large implications on plasma processes in the semi-conductor and solar cell industry. These impli-cations can affect the properties of the deposited structures either negatively or positively; while contamination of semiconductor structures with nanoparticles might have a destructive effect, the incorporation of nanocrystals in amorphous silicon layers for solar cells can result in increasing efficiency and enhanced long 1_{Note that, although the particles do not interact continuously, they still are able to interact}

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1.3 Electric fields and microparticles in the plasma sheath term stability of these devices [2]. Dust formation, growth and transport pro-cesses in silane (SiH4) discharges have been studied intensively both by means

of computer simulations [3–5] and experimentally [6–10]. More recently, the focus in the field has been shifted to hydrocarbon discharges. This increased interest has not only been triggered by applications such as those in plasma enhanced chemical vapor deposition (PECVD) and the growth of nanotubes and other nanostructures [11], but also by the negative effect of dust formation in future nuclear fusion devises. Although the extremely high temperatures in these fusion plasmas, wall material nucleates in the plasma, forming solid state dust particles and influencing the plasma operation negatively [12–14]. Also in space, dusty hydrocarbon plasmas are observed [15–17]. Plasma chemistry and powder formation and growth in hydrocarbon discharges has been investi-gated extensively by means of computer simulations [18, 19]. Experimentally, the plasma chemistry in low pressure acetylene discharges has been investigated by Benedikt [20]. Recently, particle nucleation and growth processes in low pressure hydrocarbon radiofrequency (RF) discharges have been investigated as well [6, 21–24]. Although a large amount of research has been done in this area, the dependence of dust formation on the several plasma parameters such as gas pressure and temperature is not fully understood yet. Particularly the gas temperature dependence is poorly investigated. Experimentally, a significant delay in nucleation and decreased particle growth rates have been observed in silane [9, 10] containing discharges. In 2003, Bhandarkar et al. [3] suggested in a numerical study that the delay in nucleation in SiH4 discharges is explained

by a combined effect of increasing diffusion losses due to the strong temperature dependence of the Brownian diffusion coefficient, and a reduced particle growth rate due to the decreasing plasma density in case the pressure is kept constant. However, this dependence is still not understood in detail yet and there exists a demand for more experimental data on the gas temperature dependence of dust formation in reactive plasmas.

1.3 Electric fields and microparticles in the plasma

sheath

When a quasi-neutral plasma is in contact with a solid surface, an electric space charge region – the plasma sheath – builds up near that surface due to the differ-ence in mobility between the electrons and the much heavier ions. Understand-3

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ing sheath phenomena is of major importance for almost all applications where the acceleration of ions at the border of the discharge is utilized (e.g. etching, deposition, sputtering). Even today, many processes in the plasma sheath are not fully understood since experimental data concerning plasma parameters in the sheath are extremely hard to obtain. Researchers have proposed many mod-els to predict the electric field and potential profiles within the sheath [25–27]. Experimentally, electric fields in the sheath have been determined by means of Stark splitting [28] and Stark shift [29, 30]. Another method to investigate the sheath region experimentally is based on phase-resolved probe measure-ments [31]. However, these measuremeasure-ments severely disturb the local electric field and accurate sheath models are necessary for interpreting the results. In 2005, Samarian et al. [32] introduced plasma-confined microparticles as electro-static probes in the RF plasma sheath. Having a highly negative charge, these particles are confined because the forces working on them equilibrate. Later, this method was extended for confined particles in a tailored sheath in front of an adaptive electrode [33]. The resonance of particles in the plasma sheath has been extensively studied by Zafiu et al. [34]. Until now, these experiments have all been performed at one particle position in the sheath: the position at which the forces on the particle are in equilibrium. Hence, a desired change in equilibrium position can only be achieved by either using particles with a different size, by changing the bias voltage, or by an additional ion flux [35]. These changes all severely disturb the plasma conditions in the experiment.

1.4 Objectives and structure of this thesis

According to the research areas discussed in the previous sections, the work presented in this thesis is divided in two research lines. Figure 1.1 gives an overview of the research framework.

• In the first research line, we focus on dust particle formation and growth in low pressure hydrocarbon RF discharges. By applying several diagnostics, we explore the influence of the gas temperature and the gas pressure on processes such as negative ion formation, particle coagulation and particle growth.

• In the second research line, we use microparticles under hypergravity conditions (in a centrifuge) and under microgravity conditions (during

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1.4 Objectives and structure of this thesis parabolic flights) to measure the electric field strength and the particle charge in the RF plasma sheath and in the pre-sheath.

Figure 1.1: Overview of the research framework. The research is divided in two research lines, each of them represented by two chapters discussing the research results. Below each chapter title, the studied parameters and used diagnostics are summarized.

The thesis’ structure is as follows:

Chapter 2briefly discusses dusty plasma theory relevant for further under-standing of the experiments and interpretation of the results presented in this thesis.

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Chapter 3elaborates on the several diagnostics used to study the plasmas under investigation. Discussed are the electrical characterization of the dis-charge, laser light scattering, the microwave cavity resonance technique, laser-induced photodetachment, the procedure to measure the electron temperature by means of optical emission spectroscopy, and the particle resonance method.

Chapter 4describes the used experimental setups. This section is divided in two parts. In the first part, we discuss the hydrocarbon setup used to study formation and growth of dust particles in hydrocarbon RF discharges. In the second part we present the experimental setup used to measure electric field strengths and microparticle charges in the plasma sheath and in the pre-sheath. In this part we also discuss the principles behind parabolic flights and the used centrifuge.

Chapter 5 presents and discusses the results of short time scale measure-ments exploring the first phase (negative ion formation) in the dust particle formation and growth process in acetylene-containing RF discharges. In this chapter, the time evolution of the electron density is monitored by means of the so called microwave cavity resonance technique. Additionally, this technique is combined with laser-induced photodetachment to gain more insight in the time evolution of the density of the smallest negative ions, i.e. C2H−and/or H2CC−.

Chapter 6 presents and discusses the results of long time scale measure-ments exploring dust particle coagulation and growth in acetylene- and methane-containing RF discharges. Used diagnostics are laser light scattering, electrical characterization and electron density measurements.

Chapter 7presents and discusses the results of electric field and micropar-ticle charge profile measurements throughout the plasma sheath by means of measuring the equilibrium height of confined microparticles under hypergravity conditions induced by a centrifuge. In order to obtain the desired profiles from the experimental data, a simplified sheath model is developed.

Chapter 8presents and discusses the results of electric field and micropar-ticle charge profile measurements throughout the plasma pre-sheath and the upper regions of the sheath by means of measuring the equilibrium height of confined microparticles under microgravity conditions during parabolic flights.

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1.5 Journal publications related to this thesis The sheath model of chapter 7 is extended by including the time-averaged elec-tron density and ion drag forces on the microparticle. From this simplified model, additional profiles for the electron and ion densities and for the directed ion velocity are obtained as well.

Chapter 9Finalizes this thesis with the general conclusions drawn from the performed research.

1.5 Journal publications related to this thesis

• J. Beckers, W.W. Stoffels and G.M.W. Kroesen, "Temperature depen-dence of nucleation and growth of nanoparticles in low pressure Ar/CH4

RF discharges", J. Phys. D: Appl. Phys., 42(15), 155206-1/10, 2009. • J. Beckers, T. Ockenga, M. Wolter, W.W. Stoffels, J. van Dijk, H. Kersten

and G.M.W. Kroesen, "Microparticles in a collisional RF plasma sheath under hypergravity conditions as probes for the electric field strength and the particle charge", Phys. Rev. Lett., 106(11), 115002-1/4, 2011.

• J. Beckers and G.M.W. Kroesen, "Surprising temperature dependence of the dust particle growth rate in low pressure Ar/C2H2 discharges", Appl.

Phys. Lett., 99(18), 181503-1/3, 2011.

• F.M.J.H. van de Wetering, J. Beckers and G.M.W. Kroesen, "Ethynyl anion dynamics in the first 10 milliseconds of an argon-acetylene radio-frequency plasma", Submitted to: J. Phys. D: Appl. Phys., 2011.

Bibliography

[1] I. Langmuir, “Oscillations in ionized gases,” Proc. Nat. Acad. Sci. U.S., vol. 14, no. 8, p. 628, 1928.

[2] P. Cabarrocas, S. Hamma, and Y. Poissant Proceedings of the 2nd World Conference on Photovoltaics Solar Energy Conversion, p. 355, 1998. [3] U. Bhandarkar, U. Kortshagen, and S. L. Girshick, “Numerical study of

the effect of gas temperature on the time for onset of particle nucleation in 7

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argon-silane low-pressure plasmas,” J. Phys. D: Appl. Phys., vol. 36, no. 12, p. 1399, 2003.

[4] K. De Bleecker, A. Bogaerts, and W. Goedheer, “Modeling of the formation and transport of nanoparticles in silane plasmas,” Phys. Rev. E, vol. 70, no. 5, p. 056407, 2004.

[5] K. De Bleecker, A. Bogaerts, and W. Goedheer, “Modelling of nanoparticle coagulation and transport dynamics in dusty silane discharges,” New J. Phys., vol. 8, p. 178, 2006.

[6] J. Berndt, S. Hong, E. Kovacevic, I. Stefanovic, and J. Winter, “Dust par-ticle formation in low pressure Ar/CH4 and Ar/C2H2 discharges used for

thin film deposition,” Vacuum, vol. 71, no. 3, pp. 377 – 390, 2003.

[7] A. Bouchoule and L. Boufendi, “Particulate formation and dusty plasma be-haviour in argon-silane rf discharge,” Plasma Sources Sci. Technol., vol. 2, no. 3, p. 204, 1993.

[8] A. Bouchoule, ed., Dusty plasmas, Physics, Chemistry and Technological Impacts in Plasma Processing. John Wiley and Sons, Inc., 1999.

[9] M. Sorokin, Dust particle formation in silane plasmas. PhD thesis, Eind-hoven University of Technology, 2005.

[10] W. Stoffels, M. Sorokin, and J. Remy, “Charge and charging of nanoparti-cles in a SiH4 rf-plasma,” Faraday Discuss., vol. 137, p. 115, 2008.

[11] K. Ostrikov, “Colloquium: Reactive plasmas as a versatile nanofabrication tool,” Rev. Mod. Phys., vol. 77, pp. 489–511, Jun 2005.

[12] K. Narihara, “Observation of dust particles by a laser scattering method in the jippt-iiu tokamak,” Nucl. Fusion, vol. 37, p. 1172, 1997.

[13] J. Winter and G. Gebauer, “Dust in magnetic confinement fusion devises and its impact on plasma operation,” J. Nucl. Mater., vol. 266, p. 228, 1999.

[14] J. Winter, “Dust: a new challenge in nuclear fusion research,” Phys. Plas-mas, vol. 7, p. 3862, 2000.

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1.5 Bibliography [15] P. Woods, T. Millar, E. Herbst, and A. Zijlstra, “The chemistry of

proto-planetary nebulae,” Astron. Astrophys., vol. 402, 2003.

[16] C. Keller, V. Anicich, and T. Cravens, “Model of titan’s ionosphere with detailed hydrocarbon ion chemistry,” Planet. Space Sci., vol. 46, 1998. [17] J. Greenberg and A. Li, “Tracking the organic refractory component from

interstellar dust to comets,” Adv. Space Res., vol. 24, p. 497, 1999.

[18] S. Stoykov, C. Eggs, and U. Kortshagen, “Plasma chemistry and growth of nanosized particles in a C2H2 rf discharge,” J. Phys. D: Appl. Phys.,

vol. 34, no. 14, p. 2160, 2001.

[19] K. De Bleecker, A. Bogaerts, and W. Goedheer, “Detailed modeling of hydrocarbon nanoparticle nucleation in acetylene discharges,” Phys. Rev. E, vol. 73, p. 026405, Feb 2006.

[20] J. Benedikt, “Plasma-chemical reactions: low pressure acetylene plasmas,” J. Phys. D: Appl. Phys., vol. 43, no. 4, p. 043001, 2010.

[21] S. Hong, J. Berndt, and J. Winter, “Growth precursors and dynamics of dust particle formation in the Ar/CH4 and Ar/C2H2 plasmas,” Plasma

Sources Sci. Technol., vol. 12, no. 1, p. 46, 2003.

[22] E. Kovacevic, I. Stefanovic, J. Berndt, and J. Winter, “Infrared fingerprints and periodic formation of nanoparticles in Ar/C2H2 plasmas,” J. Appl.

Phys., vol. 93, no. 5, p. 2924, 2003.

[23] H. T. Do, G. Thieme, M. Fröhlich, H. Kersten, and R. Hippler, “Ion molecule and dust particle formation in Ar/CH4, Ar/C2H2 and Ar/C3H6

radio-frequency plasmas,” Contrib. Plasma Phys., vol. 45, no. 5-6, pp. 378– 384, 2005.

[24] J. Berndt, E. Kovacevic, I. Stefanovic, O. Stepanovic, S. H. Hong, L. Boufendi, and J. Winter, “Some aspects of reactive complex plasmas,” Contrib. Plasma Phys., vol. 49, no. 3, pp. 107–133, 2009.

[25] M. Lieberman, “Dynamics of a collisional, capacitive rf sheath,” Plasma Science, IEEE Transactions on, vol. 17, pp. 338 –341, Apr. 1989.

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[26] V. A. Godyak and N. Sternberg, “Dynamic model of the electrode sheaths in symmetrically driven rf discharges,” Phys. Rev. A, vol. 42, no. 4, pp. 2299– 2312, 1990.

[27] T. Sheridan and J. Goree, “Collisional plasma sheath model,” Phys. Fluids B, vol. 3, pp. 2796–2804, 1991.

[28] U. Czarnetski, D. Luggenholscher, and H. Dobele, “Space and time resolved electric field measurements in helium and hydrogen rf-discharges,” Plasma Sources Sci. Technol., vol. 8, p. 230, 1999.

[29] C. A. Moore, G. P. Davis, and R. A. Gottscho, “Sensitive, nonintrusive, in-situ measurement of temporally and spatially resolved plasma electric fields,” Phys. Rev. Lett., vol. 52, no. 7, pp. 538–541, 1984.

[30] E. Wagenaars, M. D. Bowden, and G. M. W. Kroesen, “Measurements of electric-field strengths in ionization fronts during breakdown,” Phys. Rev. Lett., vol. 98, no. 7, p. 075002, 2007.

[31] H. Yamada and D. L. Murphree, “Electrostatic probe measurements in a collisionless plasma sheath,” Phys. Fluids, vol. 14, no. 6, pp. 1120–1126, 1971.

[32] A. Samarian and B. James, “Dust as fine electrostatic probes for plasma diagnostic,” Plasma Phys. Control. Fusion, vol. 47, pp. 629–639, 2005. [33] R. Basner, F. Sigeneger, D. Loffhagen, G. Schubert, H. Fehske, and H.

Ker-sten, “Particles as probes for conplex plasmas in front of biased surfaces,” New J Phys., vol. 11, p. 24, 2009.

[34] C. Zafiu, A. Melzer, and A. Piel, “Nonlinear resonances of particles in a dusty plasma sheath,” Phys. Rev. E, vol. 63, no. 6, p. 066403, 2001. [35] H. Kersten, R. Wiese, H. Neumann, and R. Hippler, “Interaction of ion

beams with dusty plasmas,” Plasma Phys. Control. Fusion, vol. 48, pp. 105– 113, 2006.

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2

General theory

2.1 Introduction

In this chapter, relevant theoretical considerations regarding the research pre-sented in this thesis will be discussed. In section 2.2, the basics of an RF plasma will be discussed together with the most important plasma parameters such as the electron and ion plasma frequencies, the Debye length, and the mean free path of particles moving through the discharge. In section 2.3, we will discuss the space charge region – the plasma sheath – which is created when a plasma is in contact with a surface. Here, we distinguish between a collisionless and a collision-dominated sheath by comparing the mean free path of the ions with the sheath thickness. The region coupling the plasma sheath with the bulk plasma – often referred to as the pre-sheath – and the Bohm criterion will be discussed in this section as well. Having defined these basic properties of plasma in general, we will discuss the behavior of nano- to micrometer sized dust particles in it. The most important implication of dust particles in plasma is the fact that they can become highly negatively charged. We will discuss the charging mechanism and derive expressions for the particle charge in section 2.4. All particles – both charged and uncharged – are subject to several forces. These forces will be discussed in section 2.5. In the last section (section 2.6), the formation and growth of nanoparticles in low pressure chemically reactive radiofrequency (RF) discharges is discussed.

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2.2 RF plasmas

Besides DC, AC (kHz range) and microwave (GHz range) plasmas, a commonly used plasma is the radiofrequency (RF) plasma, typically operated in the MHz range. An RF plasma is different from the DC and the AC plasmas in terms of the ability of the applied electric fields to accelerate charged particles. When the frequency of the applied voltage signal is increased, charged particles experience – due to their inertia – more and more difficulties in following the electric fields. A crucial parameter here is the plasma frequency of electrons (ωe) and that of

ions (ωi), given by ωe,i= s ne0,i0Ze,ie2 me,i0 . (2.1)

Here, ne0 and ni0 are the densities of electrons and ions respectively, having

charge numbers of Ze,i and masses of me,i. e is the elementary electron charge

and 0is the dielectric constant in vacuum. For a low pressure plasma, having a

typical electron and ion density in the order of ne0,i0≈ 1×1015m−3, we estimate

ωe/2π ≈ 284 MHz and ωi/2π ≈ 1 MHz. From this we conclude that, in an RF

plasma operated at 13.56 MHz, the inertia of only the electrons is sufficiently small to allow them to follow the applied oscillating electric field. Due to their much higher inertia, the heavier ions are not subject to acceleration by the applied RF field. These ions follow time-averaged electric fields (for example in the plasma sheath) only. The difference in physics between DC and AC plasmas on one hand, and RF plasmas on the other, becomes clear when realizing that the frequency of the applied electric fields in the DC or AC case is well below both ωe and ωi. Hence, in these cases, the ions are able to pick up energy from

the time-varying fields as well.

Another important parameter for plasmas is the Debye length. Although the plasma bulk is quasi-neutral, deviations from neutrality might occur on length scales smaller than a typical length, the Debye length λD, given by

1 λ2 D = 1 λ2 De + 1 λ2 Di . (2.2)

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2.3 The plasma sheath λDe,i = s 0kBTe,i e2_n e0,i0 , (2.3)

in which kBTeand kBTi represent the electron and ion temperature respectively.

Another important parameter is the mean free path of a specific particle under investigation, e.g. an electron, ion, atom or molecule. The mean free path λmf p,1 of particle (1) is defined as the distance it travels through the plasma

in-between two subsequent collisions with other particles (2), and given by λmf p,1=

1 n2σ12

. (2.4)

Here, n2is the density of particle (2) and σ12is the collision cross section between

the particles (1) and (2). As to be experienced in section 2.3, calculations of the mean free path of, for instance, ions can be of importance to determine whether a sheath has to be considered either collisional or collisionless. In this thesis the mean free path of ions is denoted as λmf p,i.

2.3 The plasma sheath

When plasma comes into contact with a surface, such as an electrode or a reactor wall, a positive space charge region — the plasma sheath —is created near that surface (see Fig.2.1). The principle of the development of this sheath is as follows. Initially, when the plasma is switched on, charge carriers (electrons and ions) travel towards the surface. Since the electron mass meis much lower than

the ion mass mi, the mobility µeof the electrons is much higher than the mobility

µi of the ions (µe µi) and, consequently, at this initial stage, the electron flux

Γetowards the surface is many times larger than the ion flux Γi. As a result, the

surface becomes negatively charged, obtains a negative bias voltage, and a space charge field is created between the quasi-neutral plasma bulk and the negatively charged surface. This field decelerates new electrons and accelerates ions until an equilibrium has been established. The sheath region is typically a few Debye lengths thick. In general, the plasma sheath mainly consists of ions and neutral gas atoms, while the time-averaged electron density is rather low compared to the ion density. This is because the plasma sheath is flooded with electrons from the bulk region during only a very small fraction of the RF cycle when the sum of the negative electrode bias voltage and the time-varying applied RF 13

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voltage exceeds the positive plasma potential. In the sheath of a weakly ionized low pressure laboratory plasma usually ne(z) ni(z) nN(z). Here, ne(z),

ni(z) and nN(z) represent the electron, ion and neutral densities at a given

vertical position z in the sheath respectively. Since ne(z) is low in the plasma

sheath, the excitation rate of the gas is low and, hence, in general the light emission observed from this region is much less intense than the light emission from the plasma bulk. Depending on the ratio between the sheath thickness and the mean-free path of ions, two types of sheaths can be distinguished; the collisionless and the collisional one. These two types of sheaths will be discussed in sections 2.3.1 and 2.3.2. The pre-sheath region together with the so-called Bohm criterion will be discussed in section 2.3.3.

Figure 2.1: Schematic overview of a capacitively coupled parallel plate plasma configuration in which the lower electrode is RF driven and the upper electrode is grounded.

2.3.1 The collisionless plasma sheath

For cases where the ion mean free path is much larger than the sheath thickness ξ, i.e. λmf p,i ξ, the plasma sheath is referred to as collisionless. The

colli-sionless case has been treated in many text books (e.g. in Ref. [1]) and in the last decades many collisionless sheath models have been developed. In general, these models assume quasi-neutrality at the sheath edge, Maxwellian electrons

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2.3 The plasma sheath

Figure 2.2: Schematic of the typical time-averaged electron and ion densities, and the electric field profile throughout the plasma bulk, the pre-sheath and the sheath towards the electrode. For the sake of clearness, the width of the pre-sheath is exaggerated in this drawing and will be much smaller in reality.

and cold ions. In the sheath, the electrons obey the Boltzmann’s relation1

ne(z) = ne,shexp(

eφ(z) kBTe

). (2.5)

Here, φ(z) is the position dependent potential in the sheath and ne,sh the

elec-tron density at the position of the sheath edge. The ions enter the plasma sheath with a certain velocity ui,sh and density ni,sh, and are accelerated to velocity ui

by the present static sheath electric field. Conservation of ion energy yields 1 2miu 2 i(z) = 1 2miu 2 i,sh+ eφ(z). (2.6)

Since in the sheath ionization is neglected, ion flux conservation holds

ni(z)ui(z) = ni,shui,sh. (2.7)

1_{Although within one RF cycle plasma parameters are time dependent, only time-averaged}

values are used in the derivation presented in this section.

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From Eqn.2.6 and Eqn.2.7 we find ni(z) = ni,sh(1 −

2eφ(z) miu2_i,sh

)−1/2. (2.8)

The Poisson equation reads d2φ(z) dz = − dE(z) dz = − e 0 (ni(z) − ne(z)). (2.9)

After substituting the expressions for ne(z) (Eqn.2.5) and ni(z) (Eqn.2.8) in

Eqn.2.9, and keeping in mind that ne,sh= ni,sh, the Poisson equation becomes

d2φ(z) dz = − e 0 ni,sh((1 − 2eφ(z) miu2_i,sh )−1/2− exp(eφ(z) kBTe )), (2.10) describing the potential profile through the collisionless plasma sheath.

Collisionless sheath thickness

The thickness ξ of the sheath depends on many parameters such as the electron density, the electron temperature and the sheath potential φ0. In order to derive

an expression predicting the sheath thickness, Liebermann used the Child law, originally derived for the collisionless sheath thickness of a DC discharge, and multiplied it by a factor of p50/27 to adapt for the increase in sheath thickness in the case of an RF discharge. Lieberman and Lichtenberg used the following equation for estimating ξ [1]:

ξ = r 50 27 √ 2 3 λDe( 2φ0 Te )3/4. (2.11)

2.3.2 The collision-dominated plasma sheath

For cases where λmf p,i ξ, the plasma sheath is referred to as

collision-dominated and, while traveling through the sheath, the ions experience a certain drag force due to collisions with gas neutrals. Also for the collisional case, many models have been developed to describe the sheath [2–4]. In a general fluid model, the ion motion is given by

ui(z) dui(z) dz = − e mi dφ(z) dz − nnσi−nu 2 i(z), (2.12)

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2.3 The plasma sheath in which σi−n = σi−n(ui) is the ion-neutral collision cross section which might

be a function of the ion velocity. The term on the left-hand side of Eqn.2.12 is the ion inertia term. The second term on the right-hand side represents the ion drag due to ion-neutral collisions. In the collisional sheath, the electrons still follow the same Boltzmann’s relation (Eqn.2.5) as in the collisionless case.

Collisional sheath thickness

At higher pressures, the number of ion-neutral collisions in the sheath becomes larger and the sheath becomes thinner. An analytical solution describing the collisional plasma sheath thickness ξcwas based on Sheridan and Goree [4], and

was derived by Paeva [5], yielding

ξc= 1.155 η35 u 2 5 0α 1 5 . (2.13) Here, η = eφ0 kBTe, u0= ui,sh r kB Te mi and α = λD

λmf p are the normalized potential of the wall, the normalized ion velocity at the sheath edge and the collision parameter respectively.

2.3.3 Pre-sheath and Bohm criterion

As already discussed in the beginning of this section, consistently describing the quasi-neutral plasma bulk together with the plasma sheath gives problems in governing the equations. In order to have a real solution for Eqn.2.10, the ions must enter the sheath with a minimum velocity, called the Bohm velocity uBohm, given by

ui,sh ≥ uBohm=

p

kBTe/mi. (2.14)

This relation is called the Bohm criterion. In order to have the ions accelerated up to uBohm at the sheath edge, a third region – the pre-sheath – must exist

between the plasma bulk and the sheath.

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2.4 Charge and charging of dust particles

As already discussed in section 2.3, solid surfaces in contact with plasma acquire charge due to the difference in mobility between the electrons and the much heavier ions. This charging mechanism not only applies for electrodes, antennas and probes, but also for small dust particles which may float in the discharge. Although obtaining the charge of a particle by equating the ion and electron fluxes towards the particle’s surface might look straight forward on first sight, giving good estimates of this charge is difficult and subject to intense discussions in literature over the last few decades. For example, small dust particles of only several nanometers in size can be seen as very large molecules (polymers) which – in general – can carry only one elementary charge. Discussing the charge of these small particles must be done in terms of electron attachment, ion capture and recombination processes at the particle’s surface. Hence, initially neutral nanoparticles might even become positively charged under the influence of, for instance, secondary electron emission and (photo)ionization. Especially in astrophysical clouds and interstellar space where large amounts of UV-photons and only few electrons are present, the positive charging of particles regularly occurs [6, 7]. As is to be seen in the following section, even when discussing larger particles, for which the theory of balancing electron and ion fluxes applies, other plasma relating effects make determining exact particle charges a complex task.

2.4.1 Charging processes

When a dust particle is in a plasma, it is subject to several elementary charging processes. Below we will discuss each of these briefly.

Collection of charge carriers

A solid dust particle immersed in a plasma collects plasma-created charged par-ticles (electrons and ions). Similar to the principle responsible for the formation of the plasma sheath (section 2.3), the basic charging process of dust particles in plasmas is based on the mass difference between electrons and ions. Initially the lighter and faster electrons reach the particle’s surface much quicker than the ions do, leading to a negative particle surface charge. This surface charge creates an electric field distribution around the particle which is shielded by the plasma for length scales larger than roughly λD. In the vicinity of the particle,

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2.4 Charge and charging of dust particles where the field is not yet (fully) shielded by the plasma, ions from the plasma are accelerated towards the surface, while electrons are repelled, continuing until a stationary particle charge is established.

Secondary electron emission

Highly energetic electrons and ions impinging on the particle’s surface might lead to the ejection of secondary electrons due to ionization of the particle material [6–8]. Electrons absorbed on the surface might, for instance, excite other surface electrons while their energy is absorbed. Impinging ions at low energies (below 1 keV) can become neutralized by electrons and use their energy to excite electrons from the surface. Ions with higher energy can be stopped and absorbed immediately at the particle’s surface, and the secondary electron yield can be substantially higher than unity [9]. The probability for secondary electron emission is determined by the energy of the impinging particles and by the work function of the material of the particle.

Photoemission

When photons (mainly high energy (UV) photons) are absorbed at the particle’s surface, electrons (photoelectrons) might be released from this surface [7, 8, 10]. Due to this process, small dust particles can gain positive charge. Photoelectron emission depends on the material properties of the surface material, on the surface potential and on the energy hν of the photon irradiated.

Thermionic emission

Thermionic emission is the emission of charged particles from highly heated surfaces [11]. The particle surface can be heated by, for instance, intense laser irradiation or irradiation with infrared radiation.

2.4.2 The orbital motion limited (OML) theory

The Orbital Motion Limited (OML) theory, initially developed for Langmuir probes [12], gives adequate analytical estimates for the particle charge. With this relatively simple OML theory, the charging of a spherical and isolated par-ticle with radius rp which is surrounded by a ’thick’ sheath2 can be described.

2

A thick sheath is defined as a sheath surrounding a particle and for which λD rp. 19

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The theory includes an electron density around the particle which is described by the Boltzmann distribution

ne= ne,0exp(

eφ(r) kBTe

) (2.15)

The negatively charged dust particle attracts positive ions which are collected once they have reached the particle surface. Laframboise [13] extended the original theory for a Maxwellian distribution function for ions. Here, mono-energetic ions and a collisionless plasma sheath surrounding the dust particle (lmf p,i λD) are assumed. In this case the electron and ion fluxes (Ie and Ii

respectively) to the particle’s surface are given by Ie= −πr2pne0e r 8kTe πme exp(eφ(rp) kTe ), (2.16) and Ii = πr2pni0e r 8kTi πmi (1 −eφ(rp) kTi ). (2.17)

In steady state, the particle charge does not change due to the impinging charge carriers and the sum of the electron and ion currents towards the surface equals zero

∂Q

∂t = Ie+ Ii = 0. (2.18)

By assuming ni0= ne0 in the plasma, the steady state floating potential of the

particle φ(rp) is determined by equating the electron and ion currents

exp[eφ(rp) kTe ] =r Time Temi [1 −eφ(rp) kTi ]. (2.19)

From this equation it can be observed that the floating potential of the dust particle only depends on the ratio between the ion and electron temperature and on the ratio between their masses. The dust particle charge Qp is obtained

by considering the particle as a capacitor with capacity C = 4π0rp as

Qp = Crp= 4π0rpφrp. (2.20) From the equation above it can be seen that (at constant floating potential) the particle charge varies linearly with rp. Fig.2.3 shows OML estimates of the

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2.4 Charge and charging of dust particles charge on a particle in a plasma with Te=3eV and Ti=300K as function of the

particle’s radius. As can be seen, these particles can carry a significant amount of negative charge.

Figure 2.3: OML estimate of the charge on a dust particle in a plasma with Te=3eV and

Ti=300K as function of the particle’s radius.

2.4.3 Charging time

Neglecting stochastic processes3_{, the charging of dust particles in plasma occurs}

governing

dQp

dt = Ii+ Ie. (2.21)

For Te Ti, as is commonly the case in low pressure laboratory plasmas, Boeuf

and Punset derived a characteristic charging time, given by [14]

τ = 40 e r πmi 8e pkbTi/e rpne0,i0(1 + y0) . (2.22)

3_{Stochastic fluctuations of the particle charge are small and highly frequent. Hence, a}

significant influence on the particle dynamics of micron-sized dust particles is not expected.

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Here y0 = eφ0(rp)/kbTe is the reduced potential reached for t → ∞, i.e. the

reached equilibrium value.

2.5 Forces acting on dust particles

When a dust particle is immersed in a capacitively coupled RF discharge, several forces work on it. In this section, each type of force is mentioned and discussed briefly. First, we will discuss the gravitational and the electric force. Next, we will describe the three drag forces (ion drag, neutral drag and the thermophoretic force), and finally, the radiation pressure force and mutual Coulomb interaction between particles are discussed in the last subsections.

2.5.1 Gravity

As on every mass on earth, gravity works on dust particles as well. For a spherical dust particle with radius rp and mass density ρp, the gravitational

force ~Fg acting on it is given by

~ Fg = 4 3πr 3 pρp~g, (2.23)

where ~g is the gravitational acceleration in which direction ~Fg works.

2.5.2 The electric force

In the presence of an electric field ~E, for instance in the sheath, a particle with charge Qp experiences an electric force ~FE, given by

~

FE = QpE.~ (2.24)

This means that the electric force working on a negatively charged dust particle always tends to confine the particle within the discharge.

2.5.3 The neutral drag force

The neutral drag force ~FN acts on the particle due to momentum transfer from

neutral species in the plasma to the dust particle’s surface. In a sealed plasma container, the neutral drag force is zero. However, in most industrial and lab-oratory experimental setups, continuous gas flows are used and, hence, a net

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2.5 Forces acting on dust particles transfer of momentum from the flowing gas towards the dust particle occurs. The Knudsen number, given by

Kn=

λN,mf p

rp

, (2.25)

in which λN,mf p is the mean free path of the neutrals in the discharge, appears

to be an important parameter in the analysis of neutral drag phenomena. Two regimes can be distinguished:

• For small values of KN, the thermodynamic regime prevails and ~FN is

given by

~

FN = 6πηrpv, (2.26)

in which η is the gas viscosity and v the relative velocity between the dust particle and the background gas.

• For large values of KN, the kinetic regime prevails.

In the systems under investigation in this thesis, the Knudsen number is al-ways very large and the kinetic approximation applies. We distinguish between the situation in which the relative velocity between the neutral gas particles uN and the dust particles up is small compared to the neutral thermal velocity

vth,N (

|uN−up|

vth,N 1) on one hand, and the situation in which this relative ve-locity is large compared to the thermal veve-locity on the other hand (|uN−up|

vth,N 1). For the low velocity case, when

|u_N − u_p| vth,N

1, (2.27)

Epstein determined the neutral drag force as [14, 15] ~ FN = 4 3πr 2 pmNnNvth,N(~uN − ~up). (2.28)

For the high velocity case, when

|u_N − u_p| vth,N

1, (2.29)

the neutral drag force is give by [14] ~

FN = πrp2mNnNvth,N(~uN − ~up) |~uN − ~up| . (2.30)

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2.5.4 The ion drag force

A net ion drag force ~Fi is created due to momentum transfer between positive

ions and the dust particle. Especially at the boundaries of the plasma, where positive ions are accelerated in the sheath electric field and where the ion flux is directed outwards, the total momentum of the positive ions might become significant. In contrast to the situation for the neutral drag force, the momen-tum transfer cross section is not limited to the geometrical cross section of the dust particle πr2

p. Since the positive ions Coulomb-interact with the negatively

charged dust particle, this cross section becomes significantly larger than that. The ion drag force contains two components (see Fig.2.4 for a schematical rep-resentation of both):

• The collection ion drag force ~Fi coll

is induced by the momentum transfer from the ions to the dust particle, for cases where the ions are physically collected at the particle’s surface.

• The orbit ion drag force ~Fi orb

is induced by the momentum transfer from the ions to the dust particle due to the electrostatic deflection of the ions in the potential field around the dust particle. These deflected ions create a mirror charge below the particle attracting it in downward direction.

Collection ion drag force

In Ref. [14], Boeuf and Punset derive an expression for the collection cross section σcoll for the case of mono-energetic ions depending on the ion energy

(1 2miu

2

i) and the floating potential of the dust particle under investigation

σcoll = πb2coll = πr2p[1 −

2eφ(rp)

miu2_i

]. (2.31)

With this cross section, the collection ion drag force is ~ Fi coll = nimiuiu~iπr2p[1 − 2eφ(rp) miu2_i ]. (2.32)

For a more general ion velocity distribution function, Barnes et al. [16] presented an approximate ion collection force by replacing the directed velocity ui by the

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2.5 Forces acting on dust particles

Figure 2.4: Schematical representation of the collection ion drag force on a particle induced by the physical collection of ions (left) and the orbit ion drag force induced by a positive mirror charge below the particle due to the focussing of Coulomb-deflected ions (right).

total velocity corresponding to the total ion mean energy, i.e. thermal velocity plus drift velocity

ui → vi,tot= q u2 i + vi,th2 = r u2 i + 8kBTi πmi . (2.33)

Orbit ion drag force

In order to give an estimate of the orbit ion drag force, we need to assume a potential distribution around the charged dust particle. Let us, for now, assume the cut-off Coulomb potential distribution. This distribution means a coulomb potential for rp < r < λD and zero potential for r > λD. The momentum cross

section for the orbit ion drag force is then given by [17]

σorbit= 4π

Z λD

bcoll

2pdp

1 + (p/b_π/2)2. (2.34)

Here, p is the impact parameter and bπ/2 = rp −eφ(rp)

miv2i is the asymptotic param-eter of which the asymptotic orbit angle is π/2 [17]. In Eqn.2.34, the integral 25

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is taken from bcoll to λD above which the potential of the particle is totally

shielded4_{. After integration, Eqn.2.34 turns into}

σorbit= 4πb2π/2ln( λ2_D + b2π 2 b2_coll+ b2π 2 ). (2.35)

However, Khrapak et al. [18] suggested that 900_{deflection can occur at distances}

from the particle even longer than the Debye length, and introduced

σorbit= 4πb2π/2ln( λs+ bπ/2 rp+ bπ/2 ). (2.36) Here, λs=λDeλDi/ q

λ2_De+ λ2_Di is the linearized Debye length. Using the Khra-pak cross section, the orbit ion drag force is given by

~

F_iorb= 4mivi,totni~uiπb2_π/2ln(

λs+ bπ/2

rp+ bπ/2

). (2.37)

2.5.5 The thermophoretic force

A third drag force, the thermophoretic force ~Fth, based on momentum transfer

as well, is due to a temperature gradient in the gas. The neutral gas atoms or molecules on the hot side of the dust particles have a higher thermal velocity and will transfer – when colliding with it – more momentum to the particle than those on the cold side. This results in a net momentum transfer and thus in a force on the particle in the same direction as the temperature gradient. Talbot et al. [19] derived the following expression for ~Fth

~ Fth= − 32 15 r2_p vth,N [1 + 5π 32(1 − α)]kT ~ 5TN. (2.38)

kT is the thermal conductivity of the gas, TN the neutral gas temperature and

α an accommodation coefficient which is assumed α ≈ 1 for dust and gas tem-peratures below 500 K [19].

4_{To exclude collected ions which are already taken into account in the expression for the}

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2.5 Forces acting on dust particles

2.5.6 The radiation pressure force

The radiation pressure force – sometimes referred to as the photophoretic force – is initiated when dust particles are irradiated with directed electromagnetic radiation. The fraction of radiation energy intercepted by the dust particle results into a net momentum transfer and consequently into a net force working on the particle. The radiation might be from various sources in the plasma or from outside. Radiation pressure forces induced by plasma-related radiation sources in normal laboratory plasmas are in contrast to astrophysical plasmas -not significant. However, in these laboratory plasmas, radiation pressure forces might be induced by external light sources, for instance irradiation with a strong laser beam. The radiation pressure due to irradiation with a laser beam is given by [20]

Frad =

qn1πr2pIlaser

c . (2.39)

Here, n1 is the refractive index of the medium, c the speed of light and q a

dimensionless factor. q is determined by the absorption, reflection and trans-mission of the irradiated photons on the particle, and is independent of the size of the particle and the intensity of the laser beam [20]. Ilaser represents the

laser intensity.

2.5.7 Mutual Coulomb interaction

Due to their high electric charges, dust particles in plasmas might mutually interact due to Coulomb forces. The Coulomb force Fcbetween two charges Q1

and Q2 is generally given by

Fc=

Q1Q2

4π0r2

. (2.40)

Here, r is the distance separating Q1 and Q2. In a plasma however, the

elec-trons and ions (partly) shield the electric field distribution around the charged particles. Hence, the particles do not feel each other’s full charge, but rather a reduced effective charge Qef f. For particle n, the effective charge in a

quasi-neutral plasma is given by

Qn,ef f = Qnexp(−

r λD

) (2.41)

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However, due to the high ion velocity and the electron depletion, in the plasma sheath the situation is rather complicated. Especially regarding the determina-tion of the Debye length, there exists discussion in literature. Land and Goed-heer [21] suggest to use the linearized Debye length λ−1

D =

q

1/λ2_De+ 1/λ2_Di(ui),

taking into account both the electron Debye length (λDe) and the ion Debye

length (λDi(ui)), where the latter is corrected for the ion velocity ui in the

sheath as

λ2_Di(ui) = λ2Di(vi,th)[1 +

u2_i

v2_i,th]. (2.42)

2.5.8 Dust particle confinement

As mentioned earlier, charged dust particles can be confined within a plasma. Dependent on the size of these particles, different types of forces, discussed in this chapter, become dominant and the manner in which the particles are confined becomes different. In Table 2.1 and Fig.2.5 an estimate of the differ-ent forces working on dust particles of several radii is given. Of course these forces change with the broad range of possible plasma parameters. Hence, the estimates in Table 2.1 are calculated for the following fixed set of plasma pa-rameters; nN=4.8 × 1021 m−3 (p=0.2 mbar), ne0=ni0=1 × 1015 m−3, ρp=1514

kg/m3 _{(Melamine-Formaldehyde). Furthermore, the gas is assumed to travel}

through the discharge chamber (8 cm in height) in 5 s, resulting in a gas veloc-ity of 1.6 cm/s. The temperature gradient is taken 200 K/m and the electric field 5000 V/m which is a reasonable value in the sheath. The laser intensity is estimated as the total laser output of 300 mW going through the expanded beam cross section of 13 cm2_.

From Table 2.1, it can be seen for instance, that the gravitational force – scaling with r3

p – is negligible for small particles and becomes dominant for relatively

large particles. In the following section we distinguish between the manner of confinement of micrometer-sized and nanometer-sized dust particles.

Confinement of micrometer-sized particles

In Fig.2.6, it can be seen that relatively large dust particles (micrometer-sized) become confined in the lower plasma sheath just above the lower electrode. Al-though the neutral drag force (in this case) and the ion drag force work

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down-2.5 Forces acting on dust particles

Table 2.1: Estimate of the particle charge and the different forces working on dust particles of several radii. Force rp=10nm rp=100nm rp=1µm rp=10µm Qp,OM L [e−] 4.81×101 4.81×102 4.81×103 4.81×104 Fg[N] 6.22×10−20 6.22×10−17 6.22×10−14 6.22×10−11 FE[N] 3.85×10−14 3.85×10−13 3.85×10−12 3.85×10−11 FN[N] 8.50×10−19 8.50×10−17 8.50×10−15 8.50×10−13 Fi[N] 2.73×10−17 1.87×10−15 1.04×10−13 3.32×10−12 Fth[N] 2.08×10−18 2.08×10−16 2.08×10−14 2.08×10−12 Frad[N] 4.98×10−22 4.98×10−20 4.98×10−18 4.98×10−16

Figure 2.5: Estimate of the several forces on a microparticle as function of its size.

wards while the thermophoretic force component points upwards, the gravita-tional force and the electric force are dominant in the force balance and cancel each other to confine the particles vertically at their equilibrium position.

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Figure 2.6: Schematic of micrometer-sized dust particles confined in the plasma sheath (left) and a photograph of laser-illuminated confined particles – 9.8 µm in diameter – above the RF powered electrode (right).

Figure 2.7: Schematic of nanometer-sized dust particles confined in a plasma.

Confinement of nanometer-sized particles

As discussed before, the gravitational force, which is proportional to r3

p, becomes

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2.6 Particle formation and growth balance is dominated by other forces. In Fig.2.7 it can be seen that the spatial dust particle distribution changes to a situation in which the particles can also be confined in the plasma bulk, simply because smaller electric fields appear to be sufficient to deliver the necessary confinement force.

2.6 Particle formation and growth

For the fundamental investigation of laboratory plasmas and the behavior of dust particles herein, the dust particles are often injected/introduced into the plasma. However, in many industrial processing plasmas, dust particles are not injected but grow in the discharge itself. This is because the used chemical reactive gases contain molecules that may become involved in polymerization chain reactions and form nanoparticles subsequently. Frequently used gases are silane (SiH4), Hexamethyldisiloxane (Si2O(CH3)6) and hydrocarbon gases such

as methane (CH4), ethene (C2H4) and acetylene (C2H2). Dust formation in

silane plasmas is best understood and most frequently reported in literature. Consequently, a simple four-step scenario of particle formation and growth was developed for silane containing plasmas [14]. Although details differ from gas to gas and time scales may become very different, this model applies surprisingly well for dust formation in other gases too. Since most of the powder forma-tion work in this thesis deals with acetylene discharges (and sometimes with methane), in this section we discuss the four step formation mechanism with respect to acetylene. Sometimes, however, we will elaborate on the specific case of methane. In general, dust formation and growth can be divided in four sub-sequent steps (Fig.2.8 shows a schematic representation of the different steps together with the dust particle size)

• Formation of negative ions and primary clusters: In this phase, first negative ions are formed which then polymerize with precursor molecules to form larger negative ions.

• Nucleation and cluster growth: In this phase, the clusters grow and, once reached a critical size, they turn into nanometer-sized particles. • Coagulation: In this phase, the small particles (several nm in diameter)

agglomerate rapidly to form larger particles with sizes in the order of 30 - 50 nm. Now the particles gather permanent negative charge.

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Figure 2.8: Schematical representation of the subsequent steps in the dust particle formation and growth process.

• Particle growth: In this phase, the particles grow steadily due to depo-sition of plasma created species (mainly radicals) onto their surface. In the following subsections we will elaborate on these steps briefly.

2.6.1 Formation of negative ions and primary clusters

In the first phase of the formation process of nanoparticles, chemical processes dominate. The basic principle is the triggering of polymerization chain reactions and the following up to grow to very large molecules to eventually form the primary clusters. These polymerization reactions can be initiated by either positive ions, negative ions or radicals. Although the possible lower reaction rates for the triggering by negative ions, this reaction path dominates [14]. Reason is the fact that negative ions are confined within the positive potential of the plasma and, hence, their residence time in the discharge is much larger than the residence time of radicals (subject to diffusion out of the discharge) or positive ions (accelerated out of the discharge by the electric fields in the sheath region). In acetylene-containing discharges, the dominant (and smallest)

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2.6 Particle formation and growth negative ion is formed by means of the following dissociative electron attachment (DEA) reaction onto the precursor molecule C2H2 [22]

C2H2+ e−→ C2H−+ H. (2.43)

Subsequently, the negative C2H− ion is involved in polymerization chain

re-actions with other C2H2 molecules to form larger negative ions and primary

clusters

C2H−+ C2H2 → C4H−+ H2, (2.44)

C4H−+ C2H2 → C6H−+ H2, (2.45)

C2nH−+ C2H2 → C2n+2H−+ H2. (2.46)

For the case of methane, the smallest negative ion is, just like in the acetylene case, C2H− [23]. This ion then polymerizes with CH4 to form higher negative

ions. Since the most likely way to produce C2H− is DEA onto C2H2,

acety-lene molecules must be produced in methane containing plasmas. This can be achieved by first dissociating methane by electron impact dissociation, e.g.

CH4+ e−→ CH3+ H + e−. (2.47)

The created radical then polymerizes with precursor molecules or other radicals to ultimately form C2H2.

2.6.2 Nucleation and cluster growth

The earlier discussed primary clusters grow steadily further by means of bonding with plasma created species. When a certain critical size has been reached, these clusters turn into nanometer-sized particles, i.e. they nucleate. Although these particles attain a time varying charge, most of them are neutral, and thus subject to diffusion.

2.6.3 Coagulation

Once the density of the small particles (a few nanometers in size) has reached a critical value ncin the order of 1015m−3, these particles start to agglomerate

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rapidly to form much larger and permanently negatively charged particles with sizes in the order of 30-50 nm. At the same time, the density of the particles can drop a few orders of magnitude to typically 1013 _m−3_{. From measurements}

of the particle density np and radius rp in silane discharges by Bouchoule and

Boufendi [24] and by Hollenstein et al. [25], it was estimated that the volume fraction of dust particles during the coagulation phase roughly stays constant

d dt(npr

3

p) = 0. (2.48)

In general, two models for particle coagulation are used; the neutral agglomer-ation model and the charged particle agglomeration model.

Neutral agglomeration model

In the neutral agglomeration model, the particles which move around and collide accidentally with each other experience a certain probability to stick together. The time-evolution of the particle density and radius in this phase can be given by [14]

np(t) = nc(1 + Cnct)−6/5, (2.49)

and

rp(t) = rc(1 + Cnct)2/5. (2.50)

Here, C is a constant determined by parameters such as the sticking coefficient, temperature and initial particle volume. nc and rc are the critical particle

density and radius necessary to initiate the coagulation process.

Charged particle agglomeration model

The charged particle agglomeration model includes the charging processes of the particles. Due to stochastic fluctuations of the particle charge, the nanoparticles can be charged either positive or negative. The presence of both positive and negative particles strongly enhances the agglomeration process because the col-lision of these particles is not solely due to random colcol-lisions but also stimulated by Coulomb attraction.

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2.6 Particle formation and growth Dependent on how far the coagulation process has been developed, one model fits better than the other. For example, early in the coagulation process, the cluster density is rather high and might even be a factor of 10 higher than the density of the electrons ne. Due to the charging process nedepletes significantly,

but since np> ne the time-average charge on the particle is low (closer to zero

than to one). This means that most of the time most particles are not charged at all, and that the neutral agglomeration model fits best in the beginning of the coagulation process. However, when coagulation continues, the particle density further drops and the ratio ne/np increases significant. This means that the

electron density does not deplete in such an extent as it did in the beginning of the coagulation process. Now the particles can carry more than one elementary charge and can become charged for a significant larger fraction of time. At a certain time however, the particles can become neutral or even positively charge due to the stochastic collection of free electrons and ions from the plasma. In this case, the charged particle agglomeration model would fit better.

2.6.4 Particle growth

After the coagulation phase, the particles grow steadily further by means of surface deposition of plasma created species. Although ions are deposited on the surface as well, this process is dominated by radicals (e.g. C2H) [26]. At

this stage, the particles are typically larger than 30-50 nm and carry a high permanent negative charge on their surface. As a results, the particles are confined within the positive potential of the plasma until they have reached sizes in the order of micrometers. On these relatively large particles, the balance of forces, discussed in section 2.5, alters and the particles might be lost from the discharge by either the ion drag force or gravity becoming dominant over the confining electric force. During this phase, the particles grow with rates in the order of the rates measured in thin film deposition applications. Assuming that it is most dominantly the radicals impinging on the particle’s surface that increase the particle mass mp, it can be derived that the particle growth rate

(drp/dt) is constant during the growth phase as follows; the increase in mass for

one particle is given by dmp dt = ρp dVp dt = 4 3πρp d dt(r 3 p) = 4πρprp2 drp dt . (2.51)

Here, Vp is the particle volume. The total mass per unit of time brought by the

Dust particle(s) (as) diagnostics in plasmas

Dust particle(s) (as) diagnostics in plasmas

Dust Particle(s) (as) Diagnostics in Plasmas

Dust Particle(s) (as) Diagnostics in Plasmas

PROEFSCHRIFT

Contents

1

Introduction

1.1

Plasma as the fourth state of matter

1.2

Dusty plasmas

1.3

Electric fields and microparticles in the plasma

sheath

1.4

Objectives and structure of this thesis

1.5

Journal publications related to this thesis

Bibliography

2

General theory

2.1

Introduction

2.2

RF plasmas

2.3

The plasma sheath

2.4

Charge and charging of dust particles

2.5

Forces acting on dust particles

2.6

Particle formation and growth