22nd International Symposium on Plasma Chemistry
July 5-10, 2015; Antwerp, Belgium
Numerical prediction of growth and convective transport of Si nanoparticles
around an Ar thermal plasma jet with He mixture
M. Shigeta and M. Tanaka
Joining and Welding Research Institute, Osaka University, Osaka, Japan
Abstract: Numerical simulation using the computational method, which can express fluiddynamic features of a thermal plasma flow and steep gradients of a spatial distribution of
nanoparticles, is performed to predict the effects of helium mixture on not only the plasma
behaviour but also the growth and convective transport of silicon nanoparticles.
Keywords: Thermal plasma jet, Helium mixture, Fluid dynamic instability, Nanoparticles
1. Introduction
Thermal plasmas have been expected as a powerful tool
for high-speed fabrication of nanoparticles since thermal
plasmas offer a high-temperature field with steep
gradients at their fringes where many small nanoparticles
are generated rapidly from the material vapour [1]. The
fringe of a thermal plasma flow forms eddies by fluiddynamic instability and the eddies transport the growing
nanoparticles by turbulent-like convection [2].
Meanwhile, helium gas mixture in an argon plasma
changes the temperature and flow field [3], which
consequently affects the growth and transport of
nanoparticles. In other words, the mechanism of
nanoparticle formation is controllable indirectly with
helium gas mixture. In this study, numerical simulation
using the author’s solver [2] is carried out to predict the
effects of helium mixture on not only the behaviour of a
thermal plasma flow but also the growth of silicon
nanoparticles convectively-transported around the plasma
jet.
2. Governing equations
2.1. Thermal plasma flow
Thermal plasma which is generated under atmospheric
pressure is described by the thermofluid approximation
with several typical assumptions: (i) the whole fluid
region including plasma and non-ionized gas is in a local
thermodynamic equilibrium state, (ii) the plasma is
optically thin, and (iii) the thermofluid field is twodimensional axisymmetric. The governing equations are
given as the conservations of mass, momentum and
energy:
where ρ is the density of fluid, t is the time, u is the
velocity vector, p is the pressure, η is the viscosity, I is
the unit matrix, h is the enthalpy, λ is the thermal
conductivity, C is the specific heat at constant pressure,
q rad is the radiation loss, q con is the heat generation due to
condensation, and Φ is the viscous dissipation. The
superscript tr means transposition. The momentum
exchange with nanoparticles is negligible. The
thermodynamic and transport properties which have large
variations with one or two orders of magnitude are taken
into account and obtained as temperature-dependent data
[4].
2.2. Simultaneous growth and transport of nanoparticles
The simultaneous growth and transport of nanoparticles
fabricated by thermal plasma are effectively described by
aerosol dynamics with the following assumptions: (i)
nanoparticles are liquid spheres, (ii) nanoparticles have a
monodisperse size distribution with the mean size, (iii)
electric charge effects are neglected, (iv) nanoparticle
temperature is identical to the fluid temperature, and (v)
material vapour is treated as an ideal gas. Extending our
previous model [5] treating a growth by homogeneous
nucleation, heterogeneous condensation
∂ρ
(1)
+ ∇ ⋅ (ρ u ) = 0 ,
∂t
 
∂u
2

r + r u ⋅ ∇u = −∇p + ∇ ⋅ η (∇u ) + (∇u )tr − (∇ ⋅ u )I   , (2)
∂t
3

 
and
r
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∂h
λ

+ r u ⋅ ∇h = ∇ ⋅  ∇h  − qrad + qcon + Φ ,
∂t
C

(3)
Fig. 1. Computational domain.
1
Fig. 2. Instantaneous temperature fields for:
(a) pure Ar, (b) Ar + He(10%).
and coagulation between nanoparticles, the governing
equations that also express the nanoparticles’ transport by
convection, diffusion and thermophoresis are written as
ρ
∂  np

∂t  ρ

n
 + ρu ⋅ ∇ p

 ρ




n
 = ∇ ⋅  ρD p ∇ p

 ρ







+ J − 2 2 β 0 n 11p / 6 f 1 / 6
,
np


− ∇ ⋅  K thh
∇ ln T 
ρ



 f 
f 
∂f 
ρ   + ρu ⋅ ∇  = ∇ ⋅  ρD p ∇ 
∂t  ρ 
 ρ 
ρ

+ Jg c + β 0 (nv − n s )n1p/ 3 f
Fig. 3. Statistical temperature profiles on z = 60 mm:
(a) averages, (b) normalized standard deviations.
(4)
, (5)
2/3


f
− ∇ ⋅  K thh ∇ ln T 
ρ


and
ρ
∂  nv

∂t  ρ
n

 + ρu ⋅ ∇ v
ρ


 n 

 = ∇ ⋅  ρDv ∇ v 
 ρ 


− Jg c − β 0 (nv − n s )n1p/ 3 f
, (6)
2/3
where n is the number density, D is the diffusion
coefficient, and T is the temperature. The subscripts p, v,
and s denote particle, vapour, and saturated state,
respectively. The variable f is defined as f = n p g. J is the
homogeneous nucleation rate and g c is the number of
monomers composing a nanoparticle in a critical state. β 0
is the parameter related to collision frequency given as [5]
 3vv 

 4π 
β0 = 
1/ 6
6k BTvv ,
mv
(7)
where v is the volume and m is the mass. K th is the
thermophoresis coefficient. The material properties of
2
Fig. 4. Instantaneous vorticity fields for:
(a) pure Ar, (b) Ar + He(10%).
(Line interval = 2,000 from -12,000 to 12,000 s-1)
iron are obtained from the database [6].
3. Computational conditions and method
Figure 1 shows a schematic illustration of the present
computational domain. A plasma jet of pure argon or
argon with 10% helium is ejected at 1.5 slm from the
nozzle with the radius of 4.0 mm. The temperature at the
nozzle exit is approximately 10,000 K. Silicon vapour is
supplied at 0.1 g/min with the plasma jet. The
computational domain is treated as a 2D axisymmetric
coordinate system. The governing equations are
simultaneously solved using a computational method that
the author has developed [2] to express fluid-dynamic
features of a thermal plasma flow and to capture steep
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Fig. 5. Instantaneous nanoparticle distributions for:
(a) pure Ar, (b) Ar + He(10%).
Fig. 6. Instantaneous mean diameter distributions for:
(a) pure Ar, (b) Ar + He(10%).
(Regions for n p < 1017 m-3 are cut off.)
gradients of a spatial distribution of nanoparticles.
4. Results and discussion
Figure 2 shows the instantaneous temperature fields at
the same moment. Regardless of helium mixture, the high
temperature fluid rolls up and the surrounding cold fluid
is entrained at the plasma’s fringe due to fluid-dynamic
instability, which agrees with the result observed in the
experiment [7]. Because of this convective mixing, the
plasma temperature decreases drastically. The Ar+He
plasma jet shows relatively lower temperature than the Ar
plasma jet in the downstream regions. In addition, the
temperature distribution of Ar+He plasma jet looks more
diffusive. These are attributable to larger heat capacity
and larger thermal conductivity of helium than argon.
Figure 3 shows the radial temperature profiles on z = 60
mm. In the vicinity of the jet axis (r < 7 mm), the Ar+He
plasma has lower temperature averaged with data for 102
ms, whereas the standard deviation normalized by the
local average temperature does not show much difference
between the two cases. The both plasmas exhibit large
standard deviations ranging from 0.5 to 0.7 in 7 mm < r <
24 mm. On the other hand, the Ar+He plasma shows a
large standard deviation in r > 24 mm, which means that
the region experiences much larger fluctuation of
temperature when helium is mixed with an argon plasma
jet.
Figure 4 shows the instantaneous vorticity fields
corresponding to Fig. 2. The both plasma jets exhibit
complex structures which consist of multi-scale eddies
with not only positive but also negative vorticities.
Figures 5 and 6 respectively show the instantaneous
distributions of the nanoparticle number density and the
mean diameter, corresponding to the same moment as Fig.
2. Regardless of helium mixture, a large number of small
silicon nanoparticles are found at the interface between
the high-temperature plasma and cold gas because the
silicon vapour transported with the plasma flow diffuses
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Fig. 7. Time-averaged nanoparticle profiles
on z = 60 mm: (a) number densities, (b) mean diameters.
through the interface. The generated small nanoparticles
grow up by condensation and coagulation, being
transported by the convective flow and their own
diffusion simultaneously. Therefore, the nanoparticles
farther from the jet axis tend to has larger sizes and
smaller number densities.
Figure 7 shows the time-averaged radial profiles of the
number densities and the mean diameters on z = 60 mm.
The both plasmas have the largest numbers of
nanoparticles around r = 9 mm. Meanwhile, the
nanoparticles farther from the jet axis tend to have larger
sizes, except for the pure Ar plasma in r > 24 mm. This
region showing the large difference between the pure Ar
and Ar+He plasmas corresponds to the region with the
3
large difference of the standard deviations of temperatures
as shown in Fig. 3.
5. Summary
Numerical simulation was performed using the
computational method which can express fluid-dynamic
features of a thermal plasma flow and steep gradients of a
spatial distribution of nanoparticles. The effects of helium
mixture on not only the behaviour of a thermal plasma jet
but also the growth of silicon nanoparticles convectivelytransported around the plasma were predicted numerically.
6. Acknowledgments
This work was supported by Cyberscience center of
Tohoku University for enhancement of the computational
efficiency.
7. References
[1] M. Shigeta, A.B. Murphy, J. Phys. D: Appl. Phys., 44,
174025 (2011).
[2] M. Shigeta, Abst. IUMRS-ICA2014, D2-O28-003
(2014).
[3] T. Sato, M. Shigeta, D. Kato, H. Nishiyama, Int. J.
Thermal Sci., 40, 273 (2001).
[4] M.I. Boulos, P. Fauchais, E. Pfender, Thermal
plasmas Fundamentals and Applications, 1, Plenum
Press, (1994).
[5] V.A. Nemchinsky, M. Shigeta, Model. Sim. Mater.
Sci. Eng., 20, 045017 (2012).
[6] Japan Institute of Metals, Metal Data Book, Maruzen,
(1993).
[7] E. Pfender, J. Fincke, R. Spores, Plasma Chemistry
and Plasma Processing, 11, 529 (1991).
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