Eindhoven University of Technology
BACHELOR
Determination of electric charge of nano-sized dust particles in a low pressure RF plasma
Heirman, Daan
Award date:
2019
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Determination of electric charge of nano-sized dust particles in a
low pressure RF plasma
Daan Heirman
Bachelor Final Project
Supervisors: Ir. T.J.A. Staps, Dr. Ir. J. Beckers Elementary Processes in Gas Discharges Group
University of Technology Eindhoven The Netherlands
1st of March, 2019
Summary
In this thesis, the principles of particle charging are reviewed. The goal is mea- suring the electric surface charge of nanosized Al2O3 particles (radius of 150 nm) using a novel method based on optical scattering properties of charged and uncharged particles. First an introduction of dusty plasmas is given and the charging process of dust is explained. A theoretical estimate of the surface charge is performed according to the collision-less OML theory, and is deter- mined to be −8.91 · 10−17C . A force analysis is performed to determine the location of the particles in the plasma. Secondly, a theoretical background about light scattering is explained, important scatter parameters are explained (Size parameter, index of refraction, electronic affinity) and Maxwells equations are utilized to calculate scatter efficiencies for charged and uncharged particles, depending on the electronic affinity. Using the electromagnetic boundary con- ditions at the particles, the extinction efficiency is calculated for the particles in the small particle size limit.
A literature research showed that there is a possibility for measuring charge using a charge-dependent shift in the phonon resonance at specific frequencies.
The method used to detect this shift is based on FTIR (Fourier transformed infrared spectroscopy), and could be a viable method for measuring the charge.
Contents
1 Introduction 3
2 Theory 5
2.1 Charging of particles . . . 5
2.1.1 Plasma physics . . . 5
2.1.2 Theoretical derivation of surface charge . . . 7
2.1.3 Force balance method . . . 10
2.2 Principles of light scattering. . . 12
2.2.1 Basic definitions . . . 12
2.2.2 Scatter parameters . . . 13
2.2.3 Index of refraction . . . 13
2.2.4 Derivation of scatter and absorption cross section . . . 14
2.2.5 Rayleigh Scattering . . . 15
2.3 Optical properties of charged and uncharged particles . . . 16
2.3.1 Scattering of uncharged and charged particles . . . 16
3 Experimental Set-up 19 4 Results and discussion 22 4.1 Optimizing measurement set-up: MiePlot analysis . . . 23
4.2 Spectroscopy of Argon/Oxygen mixture . . . 27
4.3 UV light as control parameter . . . 28
5 Conclusion 30
Chapter 1
Introduction
Interaction of plasma and particles is a broad area in plasma physics. A system of a plasma and particles is known as a dusty plasma. Over the past years, dusty plasmas have been investigated to understand the interaction of dust particles and plasma. For a long period of time dust particles were regarded as contamination, but over the past years useful applications have been discovered, such as powder synthesis and well-controlled surface modification [16].
Over the past years, significant advances have been made to understand interac- tion of plasma and micron-sized particles [3] However, the interaction between plasma and nano-sized particles currently isn’t well understood. The conven- tional theory for particle charging in a low pressure plasma does not suffice in laboratory environment.
One interesting feature of dust particles in particular is that scattering proper- ties of light, such as adsorption and transmittance, gets affected by the electric charge of the particles. Thereby, light can be used to measure the charge of dust particles in a plasma. Measuring the electric charge is quite important, because the charge explains the interaction between dust particles and the total charge balance.
However, measuring the charge is a very difficult task. Dust charge is influenced by a large number of factors, such as electron temperature, particle radius and density. Despite the large variety of measurement methods to overcome diffi- culties, there hasn’t been a method yet that accurately predicts the charge of a single particle. All currently known measurement methods come with advan- tage and disadvantages. Some methods are based on a force analysis [17], based on lattice wave dispersion [11], and on rotating clusters [2] all come with severe disadvantages. Some require specific and impractical laboratory set-up, very long measurement times, large measurement errors, affect the physical proper- ties of plasma or only allow measurement of micro sized particles (particles of
order 10−6nm.)
Therefore, a newer method based on R. Heinischs [10] work has been utilized.
The essence of this method is identifying the change of scattering behavior for particles due to surface charge. It turns out that for small particles (a < 1µm), the attenuation at certain wavelengths shifts linearly with the surface charge.
This method is especially interesting because it allows to measure nano-sized particles.
The goal of this thesis is to determine the viability of this method and compare it to charge prediction based on analytical models. A basic set-up (as described in chapter 4) consisting of an argon plasma and Al2O3 particles is used. This motivates the following research question: ’Is it possible to use the resonance shifts of nano-sized Al2O3 particles as charge measurement?’
In order to use this method, extensive knowledge about light scattering, plasma physics, solid state physics and electrodynamics is required, which is described in Chapter 2. This theory section is structured in three parts: i) Charging process of dust particles, ii) Scattering of uncharged and charged particles iii) Method of determining the electric charge.
Chapter 3 consists of a detailed experimental set-up of the charge measurement and equipment. Chapter 4, the most significant results are presented and com- pared to theoretical values. Furthermore, it reflects on measurement methods and results, and contains ideas to further optimize results. Chapter 6 summa- rizes the results and states whether the investigated measurement method is viable.
Chapter 2
Theory
2.1 Charging of particles
2.1.1 Plasma physics
Plasma is considered the fourth state of matter. It is an ionized gas, a gas into which sufficient energy is provided to free electrons from atoms or molecules and to allow both species, ions and electrons, to coexist. The parameter that quantifies how much gas in a certain space consists of plasma is the degree of ionization, given by:
α = ni
ni+ ng (2.1)
In equation 2.1 ni is the ion density and ngthe neutral gas density The param- eter is unity for fully ionized plasma, and approaches zero if the gas is weakly ionized.
In the absence of external perturbations, plasma can only have a net electric or magnetic field over a distance smaller than the Debye length λD, which is a characteristic parameter of the plasma: the Debye length should be much smaller than the dimensions of the plasma. Over a distance larger than λD, the randomness of electron and ion position and velocity cause the electric field and magnetic field to be zero.
The Debye length is a measure of a charge carrier’s net electrostatic effect in solution and how far its electrostatic effect persists. It’s defined as the ratio of thermal velocity of electrons (denoted as ve) and the plasma frequency (denoted as ωp:
λd= ve
ωp
= s
0kBTe
nq2 (2.2)
Figure 2.1: RF discharge [19]
0 is the electric permittivity, Te is the electron temperature, n is the density, q is the electric charge. According to this equation, the Debye length of the plasma in the set-up used in section 3 is 0.5 mm (using the values kBTe= 3 eV and n = 6.5 · 1014m−3)
To create a plasma in laboratory set-up, an electric field is applied between electrodes.
The focus will be on the plasma used in the set-up, which is a capacity coupled RF plasma (CCRF). The CCRF plasma is essentially a radio frequency plasma, typically generated by two metal electrodes separated by a small distance in a vessel, see figure 2.1[19].
Close to the wall a positively charged layer called plasma sheath can be ob- served. The thermal velocity of the electrons (eTme)12 is 100 times greater than the thermal velocity of the ions (eTMi)12, because M >> m and Te≈ Ti. In this thesis a plasma with width l and equal ion and electron densities in a vessel with grounded wall is considered, meaning Φ = 0 at the walls. Because the ion density and electron densities are equal, the net electric potential and hence electric field is 0. The fast moving electrons are not confined, and will be lost to the walls. During this small timescale ions are hardly moving, causing a net positive charge at the boundaries, as shown in figure 2.1. This net charge causes a sharp electric potential which is positive at the center of the vessel and drops quickly near the edges.
Electrons that move towards the edges are repelled back due to electrostatic force ( ~F = −e ~E), whereas positive ions are attracted to the boundaries. This effect eventually results into an electrostatic balance, such that the ion and electron flux are equal.
In a dusty plasma, the dust particles can accumulate charges, as described in section 2.1.2. Depending on plasma conditions, the particles can be charged negatively and positively.
When an electric field is applied between electrodes, the gas ionizes and becomes a plasma. These free electrons are accelerated by a RF field and ionizes the gas due to collisions, creating secondary electrons. The frequency of electrons at which electrons oscillate in the plasma is given by:
ωpe= s
e2ne
0m (2.3)
2.1.2 Theoretical derivation of surface charge
Particles embedded in a plasma acquire a net negative electric charge due to collisions with ions and electrons, such that the current of the positive and negative charges are equal.
Charging of particles in a plasma is a rather complex process. The conventional way to describe the charging of particles, namely the capturing of electrons and ions is typically given by the OML (orbital motion limited) theory. The theory essentially assumes that the effective electric potential particle decreases monotonically towards the particle. [6]
Due to additional effects, the OML theory is not an accurate description of the charging process.
process.[6]. The OML theory does not account for other effects, such as trapped ions. As a result, the OML theory over predicts the surface potential of particles.
In 2007 D’yachkov created a model that accounts for the range of collisionality of the ion motions. The model is valid from the collisioness OML regime all the way to the strongly collisional hydrodynamic regime. It states that the particle potential does not explicitly depend on the particle size, but rather on the Knudsen number:
Kn= λi
a (2.4)
λi is the mean free path of the particle, and a is the particle radius. The OML theory applies if Kn << 1, intermediate regime for Kn≈ 1 and hydrodynamic regime for Kn>> 1. To derive the surface potential, the ion current and electron current (which depend on Vp) are determined and solved for Vp.
The electron distribution satisfies the Maxwell Boltzmann equation. Utilizing the Poisson equation the electron current can be written in terms of the area,
electron density, thermal velocity and Boltzmann factor respectively.
Ie= πa2ne0
8kTe
πme
1/2
exp eVp
kTe
(2.5)
Te, me, ne0 are the electron temperature, mass, and density respectively, and Vp is the particle potential. T
Experiments have shown that the Knudsen number is not the only relevant parameter that describes the particle potential[18]. A newer model describes that the Knudsen number for the problem of particle charging needs to be defined in terms of capture radius R0, rather than the particle radius itself. The method of obtaining the equations for capture probability in terms of KnR0 is shown analytically in Varney et al. paper [20]
The probability of an ion approaching a particle colliding zero, exactly one, or more than one times is given by equations 2.6, 2.7 and 2.8 respectively.
P0= exp −1 KnR0
!
(2.6)
P1= 1
KnR0 exp −1 KnR0
!
(2.7)
P>1= 1 − (P0+ P1) (2.8)
A strong correlation between the probability of finding one collision inside the capture radius and the particle charge potential. It means that collisions change the total ion current depending on the number of collisions. When there is one collision within R0, the effect is at strongest and these ions become trapped.
The collision probabilities P0, P1, P>1are characteristic for the contributions of the OML, collision-enhanced and hydrodynamic ion transport, and therefore a formula for the entire range of Kn can be derived. [6]
Ii= P0IiOM L+ P1IiCE+ P>1IiHY (2.9) For our application,(Kn ≈ 103), IiOM L dominates. This means P0≈ 1 and the total ion current Ii can be written as:
Ii= IiOM L. (2.10)
Now, for our application the total charge on the particle is assumed to be quasi- neutral. This means the electron current reaching the particle must be equal to
the ion current. The elecron density is equal to the ion density as well, because the capture radius exceeds the Debye length.
IiOM L= Ie (2.11)
An expression for IiOM L is derived in Varney et al. [20]:
IiOM L= πa2vi,thni,0
1 − eVp
kTi
(2.12) Solving analytically for Vp, based on OML theory yields the following equation
eVp= −Te
"
ln M Ten2e mT in2i
1/2
− ln
1 − eVp
Ti
#
(2.13)
This equation is transcendental, an approximate solution is derived by Mat- soukas (1995) [17]
eVp≈ CTeln M Ten2e mT in2i
1/2
(2.14) In which C is a constant depending on plasma type. C = 0.73 for argon.
According to Klindworth et al [13], who have performed Langmuir probe mea- surements in essentially the same chamber, the electron temperature is about Te≈ 3eV the electron and ion density ne,i≈ 6.5 · 1014m−3
Solving the equation numerically for Vpyields Vp= - 0.30 V. The expected elec- tric surface charge can directly be calculated. The electric charge of a particle is given by
Q = 4πaVb (2.15)
is the electric permittivity of the medium. For Al2O3, this equals 7.88 · 10−11 F/m (farads per meter). Inserting this value and other parameters in equation 2.15 yields Q = −8.91207 · 10−17C.
The total number of electrons n is given by the total charge divided by the elementary charge of an electron:
n = Q
e (2.16)
Inserting values for Q and e = −1.602 ∗ 10−19 yields n ≈ 556 electrons.
However, the actual charge is likely much lower due to the high density of dust particles and secondary effects, such as electron emission and ion trapping. [7].
The obtained values for Q and n can be utilized as theoretical upper boundaries for the actual values.
2.1.3 Force balance method
Now that electric charge is known, it is possible to predict the orientation and size of dust clouds in a plasma, based on force balance. There are three main forces to consider: i) Gravitational force, ii) Neutral and ion drag, iii) Electro- static force Other forces, such as the radiation pressure, mutual coulomb inter- action are not considered because the order of magnitude is small compared to the other three forces. i)The gravitational force is given by:
Fg= 4
3πa3ρmg (2.17)
In which a is the radius, ρmis the particle mass density and g the gravitational constant (g=9.8m s−2) The neutral drag force is caused by the resistance of particles moving through the plasma. The momentum per unit volume equation gives the time rate of momentum per unit volume lost by neutrals [16]
fn= −Mgngvgdug (2.18)
Mg is the mass of gas particles, Ng is the density of neutrals, vgd the collision frequency and ug is the flow velocity on particles with density nd
The ion flow veloctiy is considered to be negligibly small and the neutral flow velocity is considered to be small compared to the neutral thermal velocity. The particles are assumed to be perfect hard spheres, so the collision frequency can be written as
vgd = ndσgdv¯g where σgd = πa2, and ¯vg = (8eTg/πMg)1/2. Considering the fact that the momentum lost by the neutrals is gained by the dust particles, the equation can be further simplified:
Fn= −fn/nd= Mgngπa2v¯gug (2.19) The situation for ion drift is slightly more difficult, due to Coulomb scattering.
The momentum transfer has two parts: one due to transfer when an ion is collected, and one due to Coulomb interactions of ions. Ion repulsion plays a role, the particles can no longer be approximated by hard spheres. Instead, the cross-section for collection is defined: σ = πb2c, where bcis the so called collection radius. This is determined by equating the ion current (equation 2.12) to the product of the thermal ion current flux times the area:
Ii0(1 −Vp Ti
) = 1
4eniv¯i· 4πb2c (2.20) Sustituting this equation in the cross-section yields:
bc= a
1 − Vp
Ti
1/2
(2.21)
According to Lieberman et al [16], the momentum transfer equation can be ap- proximated by a Coulomb potential that is cut off at a radius r = λD:
σ = πb20ln λ (2.22)
in which
b0= eQd
2π0M vief f2 (2.23)
Accounting for both the ion collection and Coulomb scattering, the following ion drag force is determined:
Fi= −fi/nd= M nivi,ef fui(πb2c+ πb20ln Λ) (2.24) with mn the mass number, nn the particle density, un the velocity, and νn the thermal speed.
iii)Coulomb force plays a prominent role in the interaction. The particle is assumed to be spherical and placed in a uniform electric field.
FE= QE = 4πaVbE. (2.25)
The total force on a particle (neglecting smaller forces) is the sum of the each individual force:
F~tot= ~Fg+ ~Fn+ ~Fi+ ~FE (2.26) Barkan et al [1] found methods to calculate Fi. From the calculations it be- came clear that the order of magnitude for small particles (order 10−7 nm or smaller) ensures the gravitational force is negligible compared to the ion force.
Furthermore the neutral drag force Fn is also relatively small to ion drag force.
The electric force is approximately of the same size and hence the force balance typically only consists of the electric force and the ion drag force. In the plasma bulk the ion drag force dominates, while in the plasma sheath, the electric field force dominates.
However, due to the a3 dependence of the particle size, the gravitational force does play bigger role in the force balance when particles get larger. This is particularly important if particles agglomerate. The distribution of dust parti- cles is expected to be more concentrated around the bottom of each electrode, whereas at the top of the electrode has much less large particles.
2.2 Principles of light scattering.
2.2.1 Basic definitions
Electromagnetic waves, such as light undergo scattering if they come in contact with small particles. The Mie theory is a useful tool to describe characteristic properties of materials. It explains the visual appearance of objects, such as the blue color of sky and white color of clouds, and why objects reflect or absorb light.
When light impinges on a particle, electric charges in the particle are excited into oscillatory motion. These excited charges radiate energy in form of light in all directions. This process is called scattering. A part of the incident light may be absorbed by the particle, which is called absorption.
The amount of energy per unit area per second scattered is the scattering in- tensity, and has unit W/m2. The incident intensity of radiation is represented as I0
The total energy scattered by a particle depends on the incident intensity:
Escat= CscatI0 (2.27)
In equation 2.27, Escatis the energy of the scattered light, Cscatrepresents the scatter cross section. The scatter cross section is defined as the area transverse to the relative motion within which they must meet to scatter from each other.
It is essentially the effective area that quantifies the likelihood when a beam of light strikes a particle.
Similarly, the absorption intensity can be defined in the same manner.
Eabs= CabsI0 (2.28)
In equation 2.28 Eabs and Cabs are the absorption intensity and cross section respectively
According to the conservation of energy, the loss of energy after the incident beam of light reached a particle must be caused by absorption. The total effect of scattering and absorption is called extinction. An extinction coefficient (Cext) can be defined by
Cext= Cscat+ Cabs (2.29)
Extinction can also be represented in terms of scatter and absorption efficiency.
Efficiency is the ratio between the physical cross section and geometrical crossec-
tion of the particle itself:
Q =C
A (2.30)
For spherical particles, A equals πa2
By this definition, equation 2.29 can be expressed in following terms:
Qext= Qscat+ Qabs (2.31)
The ratio of Qscatto Qext, noted as ω is called single scatter albedo. This ratio is the fraction of light extinction that is scattered, and therefore 1 − ω is the fraction of the light extinction that is absorbed.
Scattering can be categorized in two main forms, elastic and inelastic scatter- ing. When light scatters elastically, the wavelength of incident light is the same as light scattered. For inelastic scattering, the wavelength changes when light strikes a particle.
2.2.2 Scatter parameters
Elastic electromagnetic scattering can be further categorized for three different domains based on the size of the particle. The size is typically described with a dimensionless quantity called size parameter. The size parameter is defined as the circumference of the particle and wavelength of the incident light, as described in formula 2.32
ρ =2πa
λ (2.32)
a is the particle radius and λ is the wavelength of the incident light. Based on the values of the size parameter, the following three domains can be defined.
If ρ << 1, Rayleigh scattering theory applies. The particle is much smaller than the wavelength.
For ρ ≈ 1, the particle size has about the same order of magnitude as the wavelength and the Mie scattering theory applies. The Rayleigh and Mie theory will be further elaborated.
If the size parameter ρ >> 1, geometric optics apply. The size far exceeds the wavelength of the light. Scattering theory is no longer relevant in this domain.
2.2.3 Index of refraction
The size parameter, wavelength of the incident light and the index of refraction of the particle are the three fundamental parameters that determine the scatter-
ing and adsorption of light. The latter factor, index of refraction is a complex number, denoted by a capital N.
N = n + ik (2.33)
In above equation (2.33) n is the real part of the refractive index and k the imaginary part. Both the real and imaginary part depend on the wavelength λ The real and imaginary parts represent the non-absorbing and absorbing com- ponents respectively. [10]
2.2.4 Derivation of scatter and absorption cross section
In order to understand the scattering behavior, the scatter and absorption cross- sections are determined. Two scenarios are considered: neutral and charged particles.
The solution for scattering particles is obtained by solving the Laplace equation for the scalar electric potential:
∇2Φ = 0 and E = −∇Φ
Certain boundary conditions hold depending whether the particle is charged or not. Furthermore, the electronic affinity, which tells if electrons are trapped inside a particle or in a surface around it affect the boundary conditions[10].
The following 5 relations hold at the interface between a spherical particle and it’s surrounding medium, denoted by index 1 and 2 respectively:
(0E~2− 0E~1) · ~n = η0+ η (µ0H~2− µ0H~1) · ~n = 0
~n × ( ~E2− ~E1) = 0
~n × ( ~H2− ~H1) = K~ σ1E~1· ~n − ∇ · ~K = ∂η∂t
(2.34)
ηs = η0+ η is the surface charge density of the particle, in which η0 refers to the stationary charge density, and η the fluctuating charge. ~K is the surface current, and ~n is a unit vector perpendicular to the particles surface. The first four equations in 2.34 are derived from Maxwell equations, and the last one is the continuity equation. Particles are assumed ideal and satisfy the well-known linear equations for ~D and ~B: k= kE~k and ~Bk: k = µkH~k
If χ is negative, the conduction band inside the dielectric is above the potential outside the barrier. This means electrons cannot penetrate into the dielectric, and are trapped in the so called image potential in front of the surface. [8]. This image potential is a potential of a so called ’image charge’. When an electron approaches a particle, the conduction electrons of the particle screen the charge
of the particle. The screening effect is typically described as a positive image charge inside the particle.[5].
For positive electron affinity, electrons penetrate into the wall and a negative space-charge layer develops in the interior of the dielectric.
The electronic affinity therefore alters the boundary conditions for the electric and magnetic field given by equation 2.34. The surface charges may sustain a surface current, denoted by the boundary condition for the magnetic field. For χ < 0, the following boundary condition holds for the magnetic field.
ˆ
er(Hi+ Hr− Ht) =4π
c K (2.35)
in which K = σsE|| is the surface current. The equation is represented in terms of Hi, Hr, Ht: these are incident, refracting and transmitting components of the H field respectively. σs is the surface conductivity, and E|| is the parallel component of the E field. If χ > 0, there is no surface current, and therefore K in equation 2.35 equals zero.
Applying the boundary conditions leads to the following general formulas for Qscat and Qext
Qscat(m, α) = 2 α2
∞
X
k=1
(2k + 1)[|ak|2+ |bk|2] (2.36)
Qext(m, α) = 2 α2
∞
X
k=1
(2k + 1)Re[ak+ bk] (2.37)
ak and bk are coefficients which consist of spherical Bessel and Neumann func- tions, which are functions of the dimensionless surface conductivity and size parameter . For sake of clarity, ak and bk are not further elaborated here, a full derivation can be found in Klacka et al [12].
These equations serve as the basis for computational procedure of Mie theory The Mie theory can be simplified and approximated depending on the value of ρ.
2.2.5 Rayleigh Scattering
The physical understanding of scattering by small particles was established by Lord Rayleigh in the 19th century. He calculated the scattering intensity of molecules which are much smaller than the wavelength of incident light.
In order to do this, the scattering Phase function is utilized. It is the scat- ter intensity at a particular angle θ (relative to the incident beam), and then
normalized by the integral of the scatter intensity of all angles, see equation 2.38
P (θ, α, m) = F (θ, α, m) Rπ
0(θ, α, m) sin θdθ (2.38) m is the the normalized index of refraction, defined as m = N/N0 . N0 is the index of the medium surrounding the particle, which is effectively unity. ρ is the size parameter.
Several useful parameters can be derived from the phase function. It is not shown explicitly, see for a complete derivation.
The asymmetry parameter g is defined as the intensity-weighted average of the cosine of the scattered angle.
g = 1 2
Z π 0
cos θP (θ) sin θdθ (2.39)
The factor g in equation 2.39 describes how much scattered light deviates from uniform scattering in all directions. If all light is scattered forward (θ = 0) then g equals 1. Likewise, at θ = π (backward scattering), g equals -1. For a uniform scatter distribution, θ equals 0.
Another useful parameter called hemispheric backscatter ratio can be defined.
It is the fraction of scatter intensity of the light scattered backwards and light scattered in all directions, and defined by equation 2.40
b = Rπ
π/2P (θ) sin θdθ Rπ
0 P (θ) sin θdθ (2.40)
2.3 Optical properties of charged and uncharged particles
2.3.1 Scattering of uncharged and charged particles
For uncharged particles, the scattering of light is determined by the size param- eter α, the dielectric constant (ω), which depends on frequency. In general ω is complex:
(ω) = 0(ω) + i00(ω) (2.41) where 0 is the real part of the dielectric function, and 00is the complex part. If the particle does have an electric charge, light scattering is also affected by the electric conductivity of the surplus electrons. Surplus electrons can be trapped
in the particle itself, or in a layer around the particle [10]. This depends on the electronic affinity of the particle, noted as χ.
For particles with a small size parameter, i.e. ρ << 1, the formula for extinction resonance efficiency is derived according to Heinisch et al :
Qt= 12ρ(” + α” + 2τ0/ρ)
(0+ α0+ 2 − 2τ ”/ρ)2+ (” + α” + 2τ0/ρ)2 (2.42) In this equation, a single prime denotes the real parts of the dielectric constant, polarizability and surface conductivity (, α, τ respectively), and a double prime denotes the imaginary part. The excess charges enter through τ with α = 0 for χ < 0 or through α with χ > 0. The limit of Rayleigh scattering occurs at τ, α = 0 [9]. For these values, equation 2.42 simplifies to equation 2.43 (below).
The resonance is located at the wave number where
0+ α0+ 2 − 2τ00/ρ = 0 (2.43) If the particle has no electric charge, then α0 and τ00 are 0, so the resonance is at wavelength λ−10 , for which 0 = −2. Depending on the value of χ (positive or negative), the shift of resonance can be characterized in two ways. If χ < 0 then the shift is proportional to τ00, and thus to ns, the surface charge. The resonance location is then at
λ−1= λ−10 + cτns/(πcaλ−10 ) (2.44) where cτ and c are constant, denoting the change of 0 with respect to λ−1 at λ0 and the change of τ00 with respect to ns at λ0. In Heinisch et al. these equations are derived. [9]
Likewise, if χ > 0, then the shift is due to the bulk surplus charge, and the resonance is located at:
λ−1 = λ−10 − cαnb/c (2.45) In which cαdenotes the change of α0 with respect to the bulk surplus charge nb at λ0.
Figure 2.2: Position of extinction resonance depending on surface charge [9]
For Al2O3, the surface charge is plotted against wavenumber for different par- ticle sizes, see figure 2.2. The dotted line represent the solutions of equations 2.43, the solid lines are exact Mie solutions. As can be seen on the figure, the dotted and real solutions agree well with each other.
Chapter 3
Experimental Set-up
The goal of the experiment is to measure the charge using the shift of phonon- resonance of Al2O3. The methodology and set-up is similar to Kr¨ugers work [14]. The key difference is that a newer type of spectrometer is used and particles are deagglomerated in a different way.
The set-up consists of three main parts as described below: i) Injection of nano Al2O3 particles ii)Implementation of plasma chamber iii) FTIR resonance shift measurements
i) Injection of nano Al2O3 particles For optimal FTIR measurements, particles need to be (ideally) homogeneously distributed across the plasma. Homogeneous injection of Al2O3 particles is a difficult challenge, because nanoparticles stick together due to van der Waals forces. An injection method needs to deagglom- erate particles before they are brought into the plasma. The injection set-up is given below in figure 3.1
The injection valve allows to fill the reservoir with nanoparticles. During FTIR
Figure 3.1: Injection and de-agglomeration of dust particles [14]
Figure 3.2: Sketch of set-up (top), drawing of the set-up of the plasma chamber in the FTIR spectrometer (bottom)
[14]
measurements, the argon gas is brought into the reservoir using a pressure re- ducer and valve, reaching a pressure about 0.5 bar. An electric valve then allows a small amount of argon gas and nanoparticle mixture to escape the reservoir and immediately closes afterwards. The large pressure difference To further de- agglomerate the dust particles, the mixture passes through a tiny hole which has a size of 600 µm. Several methods have been performed in order to over- come this issue [14], such as dust shakers, Due to shear stress, particles could be further broken apart. Finally, particles pass through the plasma nozzle and are brought into the plasma chamber.
ii) Plasma chamber
It is needed to show that the phonon resonance can be identified from Al2O3
particles trapped in the plasma. So, in order to measure the IR spectrum, the entire chamber is placed inside a Bruker Vector 80 FTIR spectrometer (Fourier Transformed Infrared Spectrometry), which has a resolution of 4 cm−1. The set-up is shown in figure 3.2 below:
The plasma chamber has two KBr windows on opposite sides, which are trans- parent for IR light. KBr is extremely sensitive to moisture. To minimize the exposure to ambient air, and to optimize FTIR measurements, the detector and
laser source are placed close to the windows. In addition, there are two opposite glass windows and a vertical laser beam, perpendicular to the FTIR laser source to verify the shape and presence of a confined particle cloud.
iii) FTIR measurements
FTIR is used in this setup because it has great advantages over ordinary IR spectroscopy. Unlike ordinary IR spectroscopy, a FTIR spectrometer collects a large number of spectra over a wide spectral range. This greatly reduces the measurement time, and makes it possible to identify multiple resonance peaks at once. Furthermore, FTIR increases the signal-to-noise ratio, because the background noise gets reduced by the increased number of spectra (relative to the background noise)
Typically, a measurement consist of two stages: background measurement and dust measurement. A background measurement is performed when no particles are injected. Afterwards particles are injected until the confined particles are stable, a sample measurement is taken. The sample measurement contains several hundred of averaged FTIR scans and typically takes about 5 to 10 minutes each, provided that the dust particles stay stable during this time.
Chapter 4
Results and discussion
The electric charge is calculated using the OML approach and is −8.91 · 10−17 C. This means particles accumulate at most 556 electrons. The OML theory is assumed to be valid, because of the high Knudsen number (particles have a considerably large mean free path relative to the particle size). It is questionable however if it is an accurate way to determine electric charge. The ion and electron density are too high (of order 1014) to consider a collision-less regime.
This means that by following the OML theory, the ion current is underestimated, and the predicted electric charge is overestimated. Furthermore, Goree et al [7]
showed that charged dust grains in a plasma can trap positive ions in confined orbits, shielding the grain from external electromagnetic fields. Furthermore, the plasma is considered to be quasi neutral, meaning charge fluctuations can occur.
A force balance analysis showed that the particle size strongly affects the loca- tion in which particles are confined. Despite efforts to minimize the agglomer- ation (breaking up agglomerates using a large pressure gradient, and injecting through a small hole to utilize shear stress), it’s still not yet possible to get a perfectly de-agglomerated plasma. This means that there is an inhomogenous distribution of particle size (regarding clustered particles as single particles), causing an inhomogenous dust cloud in the plasma. Perhaps it is possible to use this as advantage, one could measure FTIR scans across a given area of the plasma, neglecting particles that are clustered.
The optical signatures of charged particles are investigated and compared to neutral particles. It turns out, based on Heinisch et al anomalous resonance shifts depending on particle charge are visible. For Al2O3 particles, the reso- nance shift in fig 4.1 is observed.
Based on this theory, Kr¨uger et al. developed a way to determine this shift.
Using FTIR spectroscopy, extinction resonances are identified on transmission spectra. The challenge however is that the spectrometer has a spectral resolution
Figure 4.1: Position of extinction resonance depending on surface charge ns[10]
of 4 cm−1 It is therefore necessary to adjust the conditions of the plasma in a way the net charge is increased. This leads to a more visible shift of resonance spectra and makes it possible to overcome the resolution. The next section describes possible ways to enhance the spectroscopy difference between charged and neutral particles.
4.1 Optimizing measurement set-up: MiePlot analysis
To perform optimal measurements, the scattering behavior of particles in the set-up is examined using MiePlot software.
MiePlot is a tool which uses a Bohren and Huffman algorithm [4] for Mie scat- tering off a sphere. The software calculates exact solutions of the Mie theory, and can be applied for a wide range of simulations. Two characteristics are determined: i) The Intensity versus scattering angle is modeled to determine the transmission for different angles. ii) The crossection vs wavelength is deter- mined.
Both characteristics will be determined for two cases: a) Neutral particles, b) charged particles. MiePlot does not provide an option to insert an electric charge directly, however, the scattering behavior can be modeled by indirectly imple- menting an electric charge by altering the index of refraction. The following ranges of values are used for MiePlot analysis: λ = 4000 - 10000 nm , r = 150 - 10000 nm, n = 1.79 + 1.38 i , θ = 0 - 180 degrees. The index of the medium
Figure 4.2: Intensity vs scatter angle for r = 150 nm
is approximately 1. The wavelength range is entirely in IR region, because the phonon resonance shift is expected in this region.
Secondly, clustering of particles is again investigated. No clustering at all (r is same as one particle radius) all the way up to highly clustered particles (r
= 10000 nm) are investigated. The results could provide information on how much (unavoidable) clusters of particles affect the measurement results.
i) Intensity vs scattering angle
First of all, the intensity as function of scatter angle is determined for a fixed wavelength and neutral particles (λ = 4000 nm and r = 150 nm, see fig 4.1.) As can be seen on fig 4.1, the parallel polarized (dark red) part of waves have zero intensity at θ = 90, whereas the perpendicular polarized part (bright red) appears to be independent of the scatter angle. This yields a total transmission of 12I0 for θ = 90, I0 being the intensity before the laser strikes the particle.
An optimal measurement angle is θ = 0 degrees (forward scattering) or θ = 180 degrees (back-scattering), both yield an intensity of I0, meaning no loss of intensity.
For r = 5 · 105 nm, a cluster consisting of approximately 106 particles, the scattering behavior is changed significantly, see figure 4.1
For large clusters, the average intensity (the mean of both parallel and perpen- dicular components) decreases across all angles. There is a significant difference between forward and backward scattering, meaning an inhomogeneous size dis- tribution It can be concluded that clusters do alter the optimal measurement angle. The de-agglomeration methods should be sufficient such that clusters of particles do not have an influence on Intensity vs Scatter angle measurements.
Altering the particle has no noticeable effect on this behavior ii) Crosssection vs wavelength
Figure 4.3: Intensity vs scatter angle for r = 5 · 105 nm
Figure 4.4: Extinction efficiency vs wavelength.
Secondly, the scatter crossections efficiencies as defined in equations 2.31 are determined using MiePlot. The same default values are used initially, resulting in fig 4.4
In this figure, the extinction cross section efficiency decreases monotonically as the wavelength increases. For default values, Qabs ≈ 0 across the whole region (not visible in the graph). This means Qext = Qsca. This result implies that the wavelength should be as low as possible, because for an optimal yield Qext should be as large as possible.
If the particle gains an electric charge, the complex component of the index of refraction increases. A qualitative representation is shown in figure 4.5 . The bright red, dark red and black line represent Qext, Qsca and Qabs respectively.
At an order of magnitude of k ≈ 10−3, k representing the complex part of index of refraction, (see equation 2.33), the absorption efficiency no longer becomes
Figure 4.5: Extinction vs wavelength for charged particles.
Figure 4.6: Extinction vs wavelength for uncharged particles, r = 15µm
negligibly small, and even exceeds the scatter efficiency at 5200 nm. The total scatter efficiency stays the same, but the extinction increases as result of the non-zero absorption efficiency. This means that for a shift in resonance to be as clear as possible, the electric charge should also be as high as possible.
If particle clusters occur, the scatter efficiencies have very sharp discontinuities and hence the resonance shift is inconsistent across the spectrum, as seen in figure 4.6. This further emphasizes the necessity of preventing particles from sticking together.
4.2 Spectroscopy of Argon/Oxygen mixture
In order to measure the resonance shift as described in section 2.3.2, the plasma needs to be adjusted. According to the OML theory (equation 2.14) , the electric charge increases as the electron temperature increases. This is expected, as the temperature increases, the greater the average velocity of the species, and hence it is more likely for particles to interact with free electrons in the plasma.
In order to increase the electric temperature, an Ar/O2 mixture can be con- sidered. By introducing oxygen flow towards the plasma, the oxygen becomes ionized. This results in a total of eight species: Argon atoms Ar, argon ions Ar+, oxygen atoms O, oxygen molecules O2, oxygen positive and negative ions O+, O−respectively, and electrons e−. The following particle balance equation can be defined, based on quasi-neutrality.
ne+ n−O = n+Ar+ n+O+ n+O
2 (4.1)
Oxygen is conserved, meaning:
n0O2= n0O2++ n0O2+ (4.2) Based on this, 5 continuity equations are derived in Lazzaroni et al [15] paper.
The addition of oxygen increases the electronegativity of the plasma. Elec- tronegativity is a measure of a particles ability to accept ions. It is defined as the ratio nO−/ne. The electronegativity is linked to the electron temperature, so a small increase in Te is observed if the O2 fraction increases. The electron temperature as function of plasma power is plotted for four different oxygen levels, which is seen in figure 4.7
Figure 4.7: Electron temperature as function of plasma power for four different oxygen fractions: solid (5%), dashed (10%), dotted (15%), dashed-dotted (20%)
Although oxygen has a lower ionization potential compared to argon, meaning that oxygen requires less energy for ionization, and hence, should expect a decrease of Te when the O2 fraction is increased, the electronegative character of the Ar/O2mixture explains the opposite. This means inserting oxygen into a plasma could be a viable method for increasing the electron temperature of the plasma, and thereby increasing the surface charge in particles of dusty plasmas.
4.3 UV light as control parameter
Previous section described ways of optimizing measurement result by increasing the electric charge, however, this gives no control over the total charge. It’s therefore necessary to look for a way to control the electric charge. One way to do this is introducing a UV light source to decrease the electron temperature (and thereby decreasing the electric charge)
The method is performed in Beckers et al. work. The method is based on increasing the electron density of the plasma by photo-detaching electrons from neutral Ar particles, using a UV light source with sufficient energy. This yields an additional term to equation 2.11:
Ii= Ie,i+ Ie,pd (4.3)
in which Iiis the ion current, Ie,iis the electron current before photo-detachment, Ie,pd is the contribution of photodetachment to the ion current.
By substituting non-zero photodetachment contribution, in the formula for the electrostatic potential (equation 2.13), a lower electron temperature is expected The method is convenient, because by simply altering the diameter of the UV light beam, the total electric charge is adjusted as well. It allows for a non- invasive way to alter the electric charge.
Chapter 5
Conclusion
The interaction between dust particles and the plasma is explained and calcu- lated based on OML theory.
A MiePlot analysis showed that the current set-up is viable, as long as ag- glomeration of particles is sufficiently mitigated. Measuring forward and back scattering is more efficient than other angles. Secondly, an increase of extinction efficiency is determined when particles get an electric charge.
It is theoretically possible to use resonance shift as charge measurements. The methodology in the set-up is viable, but requires adjustments. First of all, rising the electron temperature by inserting oxygen into the plasma-dust mixture increases the charge of particles, and makes measurements more viable.
Secondly, the total charge can be controlled by introducing a UV light source to control the amount of charge particles obtain. Implementing these two changes to the current set-up could allow for a non-invasive and effective method of measuring the electric charge.
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