Modeling and simulation of the agglomeration of carbonaceous dust in a radio frequency discharge using the Monte Carlo technique

DOI 10.1007/s11082-013-9694-0

Modeling and simulation of the agglomeration of

carbonaceous dust in a radio frequency discharge using the Monte Carlo technique

N. Dlimi · A. El Kebch · D. Saifaoui · A. Dezairi · M. El Mouden

Received: 12 March 2013 / Accepted: 7 May 2013

© Springer Science+Business Media New York 2013

Abstract The dust particle growth in plasmas is of major concern for safety issues in fusion reactors, and conversely has important industrial impacts. Dusty plasmas produced in lab- oratory, fusion, and in astrophysical environments have been therefore widely studied for many years to better understand the involved physical phenomena. In this work, we have investigated modeling and simulation (Ghabbouri et al. in Int Rev Phys 4(3):104–109, 2010;

Samir et al. in Chin J Phys 46(2), 2008; Louafi et al. in Int Rev Phys 6(3):297–302, 2012) a new dust-growing mechanism in capacitive radio-frequency plasma of argon/acetylene mix- ture (Ariskin et al. in Appl Phys 105:063305, 2009). Principally we studied the Brownian agglomeration in the plasma sheath by Monte Carlo simulation. We have developed a FOR- TRAN code enabling complex numeric investigations of dust particles levitating above the electrode in RF sheath. Charges, forces, balancing radii and other quantities concerning dust particles are analyzed in dependence on plasma state, position within the sheath and applied mathematical models. Commentaries and analysis of numerical results have been made.

Keywords Modeling · Plasma · Dust · Agglomeration · Brownian processes · Monte Carlo simulation

N. Dlimi·A. El Kebch·D. Saifaoui

Faculty of Sciences, Hassan II-AinChok University, BP 53 66, Maarif, Casablanca, Morocco A. Dezairi

Faculty Ben M’sik, Hassan II-Mohammadia University, Casablanca, Morocco M. El Mouden (

B

National School of Applied Sciences, El Jadida, Morocco e-mail: mahmoud.elmouden@gmail.com

1 Introduction

Plasmas full of dust make the object of intense researches since the beginning of years on 1980. They are met in several domains such as astrophysics or industry microelectronics using techniques plasma. Moreover, researches led in the context of thermonuclear fusion controlled by magnetic confinement revealed that important quantities of dusts were pro- duced within reactors with fusion. These dusts come from the erosion of walls, subjected to the fluxes of particles and warmth coming of plasma. They pose important problems of security, notably linked to the radioactivity of the tritium which they contain and can besides harm confinement of plasma and in the output of the machine. Their study par- ticularly proves to be therefore essentials, notably in the perspective of the development of the international reactor ITER. In the frame of the future International Thermonuclear Experimental Reactor (ITER), several feasibility studies were launched not only to design the reactor, but also to define the operating conditions and the security parameters. Related to this last issue, a dedicated topic deals with the ‘plasma-facing components’ and more generally about the plasma–wall interaction. High-energetic fluxes are expected in some areas, for example in the divertor zone of a tokamak where important erosion and dust pro- duction have already been observed (Federici 2001). The dust occurrence in fusion plasma induces several drawbacks. On the one hand, there are security issues: firstly, dust particles in tokamak systems may retain a large fraction of isotopic hydrogen as tritium and incidentally could spark off major safety issues. The second objective dedicates itself a the study of the mechanisms of training of dusts, and particularly agglomerates that can include sev- eral thousand particles nanometric. It is made possible thanks to the existence of waves of dusts which appear spontaneously within the clouds of dusts and provide them suffi- cient kinetic energy to gather together. The modeling and numerical simulation is much easier to consider multi-experiences mechanisms involved and assess their importance in agglomeration. The classical method to simulate agglomeration is based on the famous Smoluchowski equation (Smoluchowski 1917). In addition, the nature of aerosols involved in some industrial processes can further complicate the simulation. However, many efforts have been made to develop simulation methods (McGraw 1997; Barrett and Webb 1998;

Piskunov and Golubev 2002), which can provide accurate solutions at reasonable prices the costs. In this work, the Monte Carlo method is used to simulate the Brownian agglomera- tion process, since it has advantages over traditional discrete numerical methods, including the monitoring of the evolution of particle size, composition, morphology, charge level, and is interesting to develop the algorithm program for the simulation of systems with many variables. However, because the number of particles in a real system is too large for cur- rent computer memory, and because the accuracy of the Monte Carlo method decreases with a decrease in the number of simulated particles. The structure adopted for the devel- opment of the study is as follows: Sect. 1 is devoted to a brief introduction which contains a reminder of the plasma physics, the emergence and historical development of the theme of dusty plasmas and it applications and the particular context of thermonuclear fusion.

The Sect. 2 is dedicated to a description of the theory of agglomeration of dust using the

Smoluchowski model (Smoluchowski 1917) and creates dusty plasma in a laboratory and

also in the tokamak reactor (Ariskin et al. 2009). The Sect. 3 is dedicated to the develop-

ment of the algorithm to simulate the phenomenon of agglomeration regarded as a Brown-

ian process. To simulate this phenomenon we used the Monte Carlo technique. In Sect. 4

we give the numerical results along with discussions, analysis and commentary. This work

will culminate in a general conclusion, followed by perspectives following the study pre-

sented.

2 Phenomenon of dust growth by agglomeration of nanoparticles

2.1 Theory of agglomeration

The results of BEM (scanning electron microscopy) to identify the organization powders. Can several levels of structures: Clusters (elementary particle)-The agglomerates-Powders chain.

From these different levels of structuring, we propose a process describing the formation of powders. Smaller structures are made of carbon-hydrogen radicals. In gas phase plasma, dissociation or association of the constituents of the gas phase permit the formation of carbon- hydrogen radicals such. When these radicals saturate, they are grouped into cluster. They are generally as nearly spherical, with a diameter varying from 40 to 80 nm, the second level structuring identified. These clusters agglomerate into larger particles. Their diameters range from 100 to 400 nm.

2.2 The system used for the experimental study and dust production 2.2.1 Experiment

The reactor used in experimental studies is presented in Fig. 1 (Peng 2009; Dap et al. 2010).

It consists of a cylindrical chamber in stainless steel of diameter 36.5 cm and height 36 cm.

A removable stainless steel liner protects the walls of the chamber of dust deposits. Eight glass windows (KB7) located at mid-height are distributed around the enclosure every 45

. They can be replaced if needed by quartz windows to allow measurements of ultraviolet (UV). A further opening (DN 40) situated 5.5 cm below the upper cover can be used for instrumentation

2.2.2 Dust production technique in laboratory

Carbon dust in the reactor chamber can be produced by two distinct methods. The first one, which is quite similar to what happens at tokamak walls, Dust formation in tokamak is carbon cathode sputtering.During operation of the reactor, the walls are subjected to particle flow significant energy. Some are permanent, such as at the divertor or limiter. The materials of the walls must be very resistant to erosion and allow good heat dissipation. For example, the limiter of Tore Supra (France) is subjected a heat flow of the order of 15 MW/m

.

Fig. 1 RF discharge reactor

In this case, very small particles are obtained with low concentrations. The injection of particles dust into the core of the reactor tokamak affects plasma confinement.

The sputtering parameters are adjusted using the injected RF power in the plasma, the reactor pressure and the cathode size. The second method, which was the method we mostly used, is based on controlled injection of acetylene into the argon plasma. With this technique, C

H

dissociation produces many more carbon precursors than carbon cathode erosion.

Therefore, the amount of dust in the plasma promotes agglomeration and biggest particles and higher concentrations are observed. In this case the operating parameters are RF power, reactor pressure and acetylene flow rate. For these experiments, a stainless steel cathode was used. According to the chosen production technique, very different dust mixtures have been observed. Previous works dedicated to dust chemistry and morphology have discussed these results.

3 Modeling agglomeration

Agglomeration science is an extensive research field with numerous applications. Among several textbooks dealing with particle agglomeration, the early work of Friedlander (1977).

The evolution of the density of particles agglomeration is governed by:

∂ n

∂ t = 1 2

β

n

n

− n

i=1

β

n

(1)

In Eq. (1), the most important terms that contain all the physics of the model are the agglom- eration kernels β

. These kernels are related to the probability of agglomeration of parti- cles with characteristic volumes x

and x

. Among the various mechanisms responsible for dust agglomeration, there are Brownian and/or turbulent motion, electrostatic interactions (Piskunov and Golubev 2002), etc. The determination of agglomeration kernels of carbona- ceous dust present in the RF plasma is very tricky, since all the physics of agglomeration is not fully understood (Kortshagen and Bhandarkar 1999). During particle growth, ionic strength is F

large enough to hold small particles near the cathode sheath. The ionic strength counteracts the force of gravity F

and electric force F

which both tend to fall towards the anode powders. With the growth of particles, these two forces are increasingly important and become predominant. This implies that the particles are deposited on the anode or on the walls of the reactor as they reach a critical size

F = F

− F

− F

(2)

3.1 Kernel Brownian agglomeration

In the present study, as a first step, agglomeration is solely studied taking into account Brownian motion. Due to low pressure within the experimental reactor, the mean free path of gas components is much higher than the carbon particle diameter. Therefore, the characteristic Knudsen number K

is very large; consequently dust is not driven by gaseous flow. Therefore, according to (Dap et al. 2010), the corresponding kernel is:

β

= π

R

+ R

c

+ c

(3)

With R

being the radius of the particle and C

its mean velocity given by Max well–Boltzmann distribution as

C

=

8k

T

π m

(4)

3.2 Force

A dust particle in a plasma sheath experiences various forces (Nitter 1996). The most common are gravity and electrostatic interaction, but in some conditions other types of forces can be important, e.g. ion drag force, neutral gas friction or thermophoretic force. The charged dust particles levitate above the electrode at the position where the resulting force is zero (Fig. 2).

The gravitation force is given by

F

= −m

g = − 3

4 πr

ρ

g (5)

where m

is the mass of the particle, and ρ

is the mass density. A negative sign means that the force is oriented downwards to the lower electrode.

The electric force is given by

F

= Q

E(z) (6)

Under usual conditions the particle in the plasma sheath can be approximately considered as a spherical capacitor of the radius r

. Hence, its charge Q

and potential V

relative to the local potential of undisturbed plasma are related by

Q

= 4 π ε

r

V

(7)

The charge or potential of the particle is a result of electron and ion currents hitting its surface V

= k

T

2e ln m

T

m

T

(8) where E(z) =

U(z)

∂

z the local electric field.

The ion drag force, depending on the mechanism of momentum transfer from ions to the dust particle, consists of two components (Nitter 1996)

F

= F

+ F

(9)

Fig. 2 Diagram of the forces acting on the dust in the cathode sheath

The collection part is given by ions directly collected on the particle surface

F

= − π n

r

m

V

G

( V

) (10) The electrostatic focusing term G

defined by

G

(V

) = 1 − 2eV

m

v

if eV

≺ m

v

2 (11)

The Coulomb part is due to the momentum transfer from scattered ions not sticking to the surface

F

= − π 2 ( eV

)

m

v

n

r

ln (12)

where ln Λ is the Coulomb logarithm with

=

ε0kBTe

n_ee²r²_d

+

m_iv²_i

G

(V

) +

miv²_i

(13)

3.3 Simulation Monte Carlo algorithm

In the procedure, the N particles (initially N

) in the simulated volume are numbered from 1 to N as a particle ensemble. The simulated particles are assigned the same size distribution as the real particles. At each step of the simulation, two particles are selected to agglomerate accord- ing to the probabilities transformed from the collision rates. Gillespie’s algorithm involves calculating the cumulative probability of a particle i agglomerating with other particles in the particle ensemble in unit time, i.e. the colliding particle pair is consequently selected by the Monte Carlo approach based on the cumulative probabilities (Sheng and Shen 2006;

Friesen and Dabros 2003). Note that the agglomeration probability P

is proportional to the collision rate β

, the sum of the collision rates, S

is used instead of P

given by

S

= N

j=i+i

β

(14)

with i = 1, 2, 3, . . ., N − 1.

The time needed for the agglomeration event of the above particle pair is the inverse of the sum of all possible collision rates in the particle ensemble, given by Garcia et al. (1987)

τ = ln ( 1 − R

)

N

β

(15)

where R

is a random number uniformly distributed in the interval of 0 −1. Once particle i has been determined, its partner, particle j ( j ≥ i + 1), is selected by generating another random number R

to fulfill the following criterion:

j

l=i+1

β

≺ R

S

≺≺

j

l=i+1

β

(16)

4 Numerical simulation results and comments

With the growth of the power, the electric field near the cathode is increased, which leads the increase of the ratio E/p where p is the pressure, when the pressure is fixed. Therefore, when the power increases, the plasma is more reactive, which enhances the dissociation of the gas phase. Thus, carbon-hydrogen radicals quickly reach saturation. Particle velocity increases and the kinetic energy increases a result the formation of many elementary particles, which explains why the number of particles increases with the particle agglomeration with the target particle. In Fig. 3, we see that the average density of the agglomerated particles increases in the same direction as the RF power. This result is qualitatively in agreement with the experimental results obtained by the authors (Peng 2009).

In Fig. 4 we have shown the number of particles agglomeration phase depending on the size for different values of powers. The critical size of the dust is reduced by 400 to 250 nm with increasing the power from 20 to 100 W. A simple explanation is that the electric force

Fig. 3 Size distribution for different power (number of clusters = 20,000, pressure = 0.2 torr, primary diame- ter = 50 nm)

Fig. 4 Evolution of size according to the power (number of clusters = 20,000, pressure = 0.2 torr)

also increases with power. As a result, it alters the balance of forces acting on the powder. The maximum size of powders trapped in the discharge decreases as the electric force increases.

The training cycle is stopped and powders only small particles are formed by volume.

In Fig. 5 we have shown the number of particles agglomeration phase depending on the size for different values of powers. The evolution of the particle size distribution of C

H

can be explained by the increase in the concentration of carbon-hydrogen radicals due to the growth rate of C

H

. If the concentration is low, resulting in a very slow kinetics of particle formation. Only a small number of particles can be formed during discharge. But if the concentration of carbon-hydrogen radicals is sufficient to rapidly form many small particles. These small particles can agglomerate to changing sizes.

In Fig. 6 we have shown the number of particles agglomeration phase depending on the size for different values of powers. The critical size of the particulate matter increases from

Fig. 5 Size distribution for different number of dust (Power = 20 W, pressure = 0.2 torr, primary diame- ter = 50 nm)

Fig. 6 Evolution of the size depending on the number of dust (Power = 20 W, pressure = 0.2 torr)

200 to 400 nm, with the increase in the density of carbon-hydrogen radicals from 10

to 2 . 5 × 10

, these result is in agreement with the experimental result (Peng 2009). A simple explanation is that the average time of free flight and reduces the increase in the probability of attachment of clusters.

In Fig. 7 we have shown the number of particles agglomeration phase depending on the size for different values of powers. When the pressure increases, the collision frequency between electrons and neutrals also increases, it leads to a reduction of the electron temperature. With decreasing electron temperature, dissociation and ionization are increasingly low, which limits the formation of particles. This explanation is in good agreement with observations, which show that the amount of dust decreases as the pressure increases.

In Fig. 8 we have shown the number of particles agglomeration phase depending on the size for different values of powers. We also find that the average size of dust increases with increasing pressure from 200 to 600 mTorr. In this range, we can assume that the dust grow with the increase in the frequency of collisions between particles and radicals carbon- hydrogen, which means also reducing the amount of dust elementary.

Fig. 7 Size distribution for different pressure (number of clusters = 25,000, Power = 20 W, primary diame- ter = 50 nm)

Fig. 8 Evolution of size according to the pressure (number of clusters = 25,000, Power = 20 W)

5 Conclusion

The present study was dedicated to the understanding of dust agglomeration cycles in RF plasma discharges. In order to better understand experimental results, the numerical model are developed using the Monte Carlo technique simulation. A FORTRAN code was developed to study the dynamics of the agglomeration. In our simulations we use the same parameters as the experiment. The shape of the distribution curve of the agglomerated particles is in qualitative agreement with the experimental curve authors’ results (Peng 2009).Also we have showing that the average agglomerated particle size is the same order of magnitude as that found by experiment. In perspective we plan to develop this work with the fluid model to study the agglomeration dynamics.

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