Study by Inverse Method the Size Distribution of the Particle of Carbonaceous Generated in RF Discharge

One of the difficulties encountered in the study of dusty plasmas is related to the knowledge of the size of the dust particles present. A variety of sources, physical and chemical mechanisms of formation, causes a wide variety of sizes and morphologies of dust. The diameter of a dust will not be unique but spread over several orders of magnitude. Its distribution in number, surface, mass or volume is called distribution. It is important to know this distribution in particle size because it strongly impacts the physical and radiative processes. To have a dust distribution in situ is very difficult; the reverse method can identify the particle populations from light extinction measures. In this study, we present an inversion procedure with a Tikhonov regularization dedicated to the determination of volume size distribution (V-PSD) from extinction measurements corresponding to the different wavelengths obtained by the Extinction Spectrometry technique.


Introduction
Dusty plasmas are the objects of intense research since the beginning of the years 1980. They are met in several fields such as astrophysics [1] [2] or Mi-How to cite this paper: El Kebch, A., El Mouden, M., Dlimi, N. and Saifaoui, D. cro-electronics industry using the processes plasma [3]. In addition, the researches undertaken in the context of the thermonuclear fusion controlled by magnetic containments reveal that important quantities of dust are produced within the plasma reactors. The formation of these dust results from the erosion of the plasma reactor walls by physical sputtering [4], by chemical erosion or by melting/sublimation consecutive to an abnormal event. These dusts can pollute the confined plasma and play a critical role in the effectiveness of the plasma discharge by causing significant losses by radiation. In addition, these dusts can be a health problem especially when they contain tritium [5] [6] [7].
Due to the importance of these topics, in particular in the framework of the development of the international reactor ITER, many experimental works have been performed for many years to understand the growth mechanisms and the associated dynamics of these dusts [8]. To study these two last variables, the measurements of the size distribution and the concentration of particles flowing in dusty reactive plasma are really relevant. But for that, these measurements need to be non-intrusive, realized within the plasma reactor as well as being spatially and temporally resolved. This is why the technique referenced as "Light Extinction Spectroscopy (LES)" seems to be the most appropriate for dust assessments in fusion device reactors.
An incident radiation that propagates in a vacuum, undergoes no modification either in its intensity or its state of polarization [9]. An electromagnetic wave that propagates in a medium of electric charges will cause the oscillation of the charges, so a part of the energy of the wave is absorbed as an oscillating electric charge radiates. This absorbed energy is re-put into the medium in the form of an electromagnetic wave propagating in a new direction.
This diffusion results in [10]:  Modification of the direction of propagation of the incident wave;  The absorption that will characterize the decrease of the energy of the incident wave. It is therefore interesting to note that the diffusion phenomenon is related to the nature of the propagation medium and depends on the geometry of the particles of the medium.
Due to the importance of the subject dusty plasma and to understand the mechanisms of growth of these dusts, as well as their dynamics, many experimental works have been carried out for many years. The analysis and the knowledge of the particle size of the dust in the plasma make it possible to better understand the behavior of the dust.
These small particles have been of particular scientific interest, Given the wide variety of applications in materials technologies, and in the electronics and optics field. A promising technique for accurate real-time and in-situ measurement of the size and concentration of sets of nanoparticles is the extinction spectroscopy (TES) technique. This technique is based on measuring the transmission spectrum of a light that passes through a medium that contains particles (liquid or solid). The intensity of this transmitted light is a function of its wavelength, the size, the concentration and the refractive index of the particles. The spectrum of transmitted light is modeled using a light scattering theory, and a data inversion algorithm is used to determine the volume size distribution of the particles. The measurement technique has several advantages including limited optical access, the ability to perform long-range measurements, a simple optical configuration and a short measurement time. These advantages could make this technique widely used in industrial and scientific environments.
Unfortunately, the data inversion algorithm requires the resolution of a Fredholm equation of the first type, which is a well-known but poor resolution problem. This problem must be regularized to ensure that the determined size distribution has a physical meaning. A detailed literature exists regarding this technique. [11] [12] [13] [14].
In the work reported here, LES is used to characterize the growth of carbonaceous particles produced in an Argon RF discharge. After this short introduction, the experimental setups are described: the plasma reactor, the LES device as well as its principles and the inversion and regularization procedures developed to interpret the recorded experimental data. In the last part, experimental results are presented and discuted.

Experimental Plasma Reactor [15]
The reactor used in this experimental study is shown in Figure 1. equipped with a dust filter. A mixture of argon and acetylene (C 2 H 2 ), whose quantities are controlled using mass flow meters, is injected within the chamber through micro-holes. In the present study, the mixture fraction in volume is fixed to Y C2H2/Ar ≈ 43%, (Ar 7 sccm + C 2 H 2 3 sccm) while the static pressure of Argon and the electrical power are maintained respectively at P = 2.7610 −1 mBar and P elec = 20 W.
Because of its geometry and its operating mode, a large amount of carbonaceous particles, whose diameters range from tens to hundreds of nanometers, are trapped near the cathode. This is the result of a balance between several forces: the gravity force, electrostatic forces and the drag forces.
A. El Kebch et al. This is this cloud of particles that is studied hereby light extinction spectroscopy (LES) to determine the particle size distributions (PSD) and the particle concentration near the cathode during the plasma ignition.

LES Principle
The LES technique consists in illuminating the dust to characterize by a collimated beam of polychromatic light [16]. If the collection of multi-scattered photons is negligible, the spectral transmission rate of the incident beam can be written: where m  is the complex refractive index of each particle.

Inversion Procedure
From a mathematical point of view, the Equation (2) is an inhomogeneous Fredholm equation of the first kind. Our problem is to determine the concentra- where V(D) is the particle size distribution in volume and v(D) the normalized particle size distribution in volume. If the number quantities are replaced by volume ones and by introducing the constant Λ = −3/2L, the Equation (2) takes the form: The previous integral equation can be discretized as follows [16]: where V(D) is the quantity we want to determine and that can be discretized by the vector V j with 1, 2 , j M =  , (S ij ) is the element of the "extinction matrix" whose dimensions are equal to N × M. The latter is determined for a given wa- The measured transmission spectrum can also be written as a one-dimensional vector T whose each elements T i represents the measured transmission for the wavelength λ i within a particles cloud. To determine the discretized PSD in volume V, we just have to solve the algebraic equation:

SV T
≡ . This equation admits a trivial solution: However to stabilize the inversion procedure, it is preferable to iteratively minimize the quantity SV-T with a least-square algorithm and a non negative constraint: The minimization procedure can be realized with orthogonal algorithms [19].

Regularization of the Problem
It has to be noted that the previous method has already been used successfully to inverse critical scattering patterns produced by clouds of bubbles. But in the argon-acetylene RF discharge studied here, the particles generated have a diameter where H is a smoothing matrix that is here the identity matrix, γ a regularization parameter whose optimal value is estimated by the method of L-Curve [21] [22], V 0 an assumed initial solution vector which is here chosen equal to zero. The latter is connected to a PC which permits to control it, to record data or to inverse them.

LES Measurement Protocols
To perform LES measurements, the following experimental protocol has been used: 1) ∆t = 0 s, the LES lamp is ignited during ∆t = 120 s to stabilize its intensity, 2) ∆t = 120 s, a plasma made of pure argon is ignited during ∆t = 120 s, These five phases will be discussed later in the section «Effects of the distance relative to the cathode». Figure 2 presents the temporal evolution of the transmission recorded for three different vertical positions of the LES probing beam below the cathode: (a) ∆h 1 = 0 cm, ∆h 2 = 0.5 cm, ∆h 3 = 1 cm. In these three cases, the plasma operating conditions are given in Section 2.1.

Experimental LES Measurements
Several commentaries can be made regarding the Figure 2: • Whatever the wavelength for a given position, the temporal evolution of the transmission spectrum has the same shape. As already shown on Figure 2, the transmission rate is lower as the wavelength is low.
• More the LES measurements are made close to the cathode, more the transmission rate decreases. For example, for a wavelength of λ = 300 nm, the transmission rate decreases from 70% at ∆h 3 , to 50% at ∆h 2 , to fall to 15% near the cathode (at ∆h 1 ).
When the plasma is ignited and C 2 H 2 is injected, i.e. during the phase III, we can notice that near the cathode (at ∆h 1 ) and for a given wavelength, the temporal evolution of T present several variations. Conversely far away from the cathode (at ∆h 3 ), T remains almost constant over the phase III, except during the first 100 s of this phase. It seems that the temporal variations of T for a given wavelength are attenuated when the distance between the probed volume and the cathode increases. This may result from the movements of carbonaceous particles produces in this area.

Inversion Parameters
The space of the diameters particles is discretized in M = 100c lasses of 10 nm wide, distributed linearly between D min = 10 nm and D max = 1 µm. The spectral discretization is effected using J = 71 wavelengths distributed between λ min = 300 nm and λ min = 1000 nm. The length of the probe volume is equal to L = 30 cm, i.e. the reactor diameter. Figure 3 shows the PSD in (noted V-PSD) volume obtained by LES measurements at the 3 positions described before and with the plasma operating conditions described in Section 2.1. Figure 3 indicates that near the cathode the particle size distribution is broader, it is observed that when approaching the cathode increases the density of the dusty, it is deduced that the powders are trapped near the cathode and are deposited on it during the discharge which is why the greater the distance of the cathode, more transmission rate approaches 1 (whatever the wavelength), more the density of the dust decreases.    Figure 4 reports the temporal evolution of the mean diameter measured by LES value oscillates between ∆t = 360 s and ∆t = 660 s and lastly increases after ∆t = 660 s of C 2 H 2 injection. This evolution of D v is in good agreement with the temporal evolution of the transmission reported in the section «Experimental LES measurements». In addition, Figure 4 shows that the temporal evolution of the concentration in volume C v follows the one of D v . This reveals that particles, which are produced near the cathode, move towards the anode between ∆t = 360 s and ∆t = 660 s after the start of the C 2 H 2 injection. Then C v increases as well as D v increases or oscillates. This may reveal the formation of a new generation of podwer.

Effects of the Optical Index on the Granulometry
The three main drawbacks of using the LES technique to study the growth of carbonaceous nano-powders produced in RF discharges are that 1) the composition of theses nano-powders is not known, 2) in addition their composition can vary during the plasma ignition, 3) lastly the inversion procedure of LES measurements requires to know the refractive index on a large spectral range. In a first approach, different refractive indices found in the literature can be used.
But the effect of the choice of a refractive index over another on the LES inversion results must be estimated. Table 1 summarizes the 3 different refractive indices of carbonaceous particles tested in this work. As might be expected, the choice of the refractive index used to inverse LES measurement modifies the reconstructed PSD (see Figure 5). The associated concentration in volume is reported on Table 2. It appears that less the material constituting the nano-podwers is absorbent, more the concentration in volume is higher.   Table 1.

Conclusions
We have studied the in-situ particle size distribution of the dust obtained by a low pressure Argon acetylene mixture RF discharge, using a commonly known method: inverse method accompanied by a regularization of Tikhonov which allows us to find with a good precision the distributions in particle size responsible for extinguishing the light beam used as a non-intrusive diagnostic means.
The appearance of small particles, generally less than 100 nm, is frequently observed in the particle size distributions found by the reverse method. These particles should be ignored when using the results.
The reverse method has been applied to experimental data. This allowed on In this case the results provided by the inverse method depend strongly on the choice of optical indices used. Despite this, the trends observed during the evolution of the particle population are qualitatively similar. The choice of optical index 3rd case of Table 2 seems to provide particle sizes close to those found in the literature or well by experimental measurements using Scanning Electron Microscope (MEB).
In order to obtain information on the particle size distribution of the nanoscale dust cloud, we have chosen a technique of inverse computation by using experimental measurements of the transmittance obtained by a spectrometer and to be based on the hypothesis of spherical particles (theory of Mie). We found that the PSD-V is the lowest closest to the cathode, which indicates that the density of powders is greater when approaching the cathode; this shows that the powders remain in the vicinity immediate result of the cathode, and in good agreement with the measurements of the transmittance obtained in the spectral range of between 300 nm and 1000 nm.
Since we do not have a precise knowledge of the optical indices of dust, we have studied the influence of the choice of optical indices used on the results; in this sense the inversion procedure has been applied to experimental data using the radiative properties calculated using 3 sources of different optical indices; the result that we have deduced is that the less the material is composed of absorbing particles, the lower the volume fraction.
The permanent interaction between the digital aspect and the experimental aspect will eventually lead to a more complete understanding of the agglomeration phenomenon. Possessing a tool to quantitatively monitor the size distribution of powders from non-intrusive optical diagnostics and easily implantable in a reduced environment is an important feature for a better implementation on tokamaks, including the future ITER reactor.

Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.