SENAC Vol. 47, No. 4（2014. 10）— 11 —［共同研究成果］

Computational prediction of niobium-silicide nanopowder growthin thermal plasma synthesis
Masaya Shigeta
Joining and Welding Research Institute, Osaka University
A growth process of niobium-silicide nanopowder from the precursory binary vapors of niobium andsilicon in a thermal plasma environment has been computed using a unique mathematical model as a recent
application of supercomputing. The collective and simultaneous growth behavior through nucleation,
co-condensation and coagulation has been clarified. Because of the time lag of binary co-condensation
between niobium and silicon, the synthesized nanopowder has wide ranges of the size and the silicon
content.

1.

Introduction

Efficient production of nanopowder which is composed of nanoparticles is intensively expected fornumerous applications in industrial, biomedical, and environmental purification processes because
nanoparticles exhibit unique electronic, optical, and catalytic properties which are different from those oflarger particles of micrometer size or bulk materials. However, the synthesis of functional nanoparticles ofceramics and alloys seems to be practically arduous by conventional methods since their raw materials
have too high melting points to be decomposed. Even combustion processes cannot generate a sufficiently
high temperature to vaporize the raw materials; furthermore, combustion requires an oxidation atmosphere,
which is undesirable for synthesis of nanoparticles and which produces contamination from combustion
products (CO2, H2O, etc.).
To overcome such a problem, thermal plasmas have been anticipated as a promising tool for efficientfabrication of nanoparticles [1] because thermal plasmas offer a distinctive thermofluid field involving ahigher temperature, high chemical reactivity and variable properties. Moreover, thermal plasmas arecontrollable by external electromagnetic fields [2-4]. By virtue of such a field, a large amount of raw
materials, even that with high melting/boiling points, are vaporized completely. The vapors of the raw
materials are transported downstream with the plasma flow to the plasma's tail region where thetemperature decreases rapidly; and consequently the vapors become highly supersaturated. Because thesupersaturated state is unstable, the vapors change their phases quickly to very small particles through thecollective and simultaneous formation through nucleation, condensation and coagulation among theparticles themselves. In consequence, nanoparticles are mass-produced at a high rate.
However, it is still difficult to investigate the formation mechanism of functional nanoparticles ofceramics and alloys generated in/around a plasma because the process involves remarkably severe andintricate heat/mass transfer associated with phase conversions in a few tens of milliseconds in a complex
thermofluid field. Experimental approaches using a direct measurement or observation are currently
impossible owing to technological limitations, whereas numerical studies have struggled with themathematical formulation to express the intricate multi-scale physics and the shortage of computational
resources to obtain practically meaningful solutions. Meanwhile, the recent progress of supercomputing
systems is noticeable, which has made it possible to simulate multi-scale and complicated phenomena with
sufficient accuracy: even for numerical investigations of the nano-material processing by thermal plasma.
This paper presents a computational clarification of a binary growth process of a functional ceramic
nanopowder under a thermal plasma condition.

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2.

SENAC Vol. 47, No. 4（2014. 10）

Model description

The vapor phase syntheses of a binary functional nanopowder can be computed using a uniquemathematical model that was recently developed by the author [5, 6]. The model describes a collective andsimultaneous growth process of two-component nanoparticles in a binary vapor system with the following
assumptions:(i) nanoparticles are spherical;(ii) inertia of nanoparticle is negligible;(iii) the temperature of nanoparticles is identical to that of the bulk gas surrounding them;(iv) heat generated by condensation and the electric charge of nanoparticles are neglected; and(vi) the material vapors are regarded as an ideal gas.
To treat the particle size and composition during the growth, the model introduces the particle
size-composition distribution (PSCD) on the basis of two-directional nodal discretization [7]. Using thePSCD, the net production rate of nanoparticles having the volume vk and the content of the second material
xn is written by the increment of the number density N during the infinitesimal time t as:

N k ,nt J binary k( nucl) n( nucl) il) i(,cond  i ,k  l ,n N i.kl ,n( cond )i ,l ,kt(1)1) ( coag) i(,coag
j ,k  i , j ,l ,m ,n  i , j ,l , m N i .l N j ,m  N k ,n   i ,k ,l , n N i .l
2 i j l milHere, i and  i denote the splitting operators for the size and composition, respectively.  is theKronecker's delta. The first term on the right-hand side represents the contributions of binary homogeneousnucleation. The second term means the production rate caused by vapor condensation on the particles
having vi and xl. The third and fourth terms express the gain and loss by coagulation among nanoparticles.
 is the collision frequency function for nanoparticles resulting from Brownian motion [8]. Subscripts i andj denote the node numbers for size, whereas subscripts l and m signify those for composition. To obtain thehomogeneous nucleation rate Jbinary for a binary system, the theoretical formula derived by Wyslouzil andWilemski [9] is used.
When the growth rate of nanoparticles by heterogeneous condensation of the vapor of material M, thefollowing formula considering the rarefied gas effect correction is used to calculate the volume incrementv(M)i,l during the infinitesimal time increment t:v( M )i ,lt 2d i Dvap( M ) vvap( M ) N vap( M )  N S ( M )i ,l 0.75 ( M ) (1  Kn i )2 0.75 ( M )  0.283 ( M ) Kn i  Kn i  Kn i (2)where, d is the diameter, D is the diffusion coefficient, and Kn is the Knudsen number. The subscript vap
denotes vapor.  represents the accommodation coefficient which has a value of 0.1 here. NS means thesaturated vapor concentration considering the effects of material mixture and surface curvature [10]. Notethat the nanoparticles are allowed to grow by condensation only when the free energy gradients for particle
formation, W, in a binary system is negative or zero:Computational prediction of niobium-silicide nanopowder growthin thermal plasma synthesis

W0n( M )— 13 —(3)where, n(M) represents the number of monomers of material M in a nanoparticle. W is composed of thechemical potentials and the surface energy. The population balance equations of the material vapors arealso computed simultaneously because the number densities of the material vapors crucially affect thegrowth process:N vap ( M )N i.l v( M ) i ,l  J binary  k( nucl ) n(*M )  t
vvap ( M ) tkil(4)where n*(M) represents the number of monomers of material M composing a stable nucleus. In thecomputation, the melting point depressions due to nano-scale size and the mixture effect are considered[11]. It is assumed that the nanoparticles with the temperature lower than their melting point cannot growby coagulation.

3.

Computational conditions

The synthesis of niobium silicide nanopowder is selected as a target process. Those nanoparticles haveattracted a plenty of interests of physicists and chemists as well as engineers because their collective andsimultaneous growth from the vapor phases is a complicatedly interacting process and the fabrication ofthem with well-controlled sizes and compositions is still very difficult. Those problems are fundamentally
attributed to the large difference of the saturation vapor pressures of niobium and silicon. Currently, only
the model described in the previous section seems to be able to analyze the detailed mechanism of thecollective and simultaneous growth of niobium silicide nanopowder. The computation is demonstrated
under a typical condition of the synthesis using a thermal plasma as follows. Coarse precursory powders of
niobium and silicon are injected in the ratio of Nb:Si = 1:1 into the thermal plasma. The vapor ratio of theprecursors to Ar is set to be 0.5 %. They are vaporized immediately in the high-enthalpy plasma and thevapors are transported downstream with the decrease in their temperature. The computation begins fromthis situation in which the vapors are about to be supersaturated. The temperature decreases monotonicallyat the cooling rate of 105 K/s. The mathematical model embeds the discrete nodes in the size-composition
space. Here, 101 x 101 nodes were placed in the size and composition directions, respectively. The
material properties of niobium and silicon were obtained from Ref. [12].

4. Results and discussion
Figure 1 shows the evolution of the PSCD in the Nb-Si binary system, which expresses the collective
and simultaneous growth of niobium silicide nanopowder. As the temperature decreases, niobium vapor
becomes supersaturated at a higher temperature than silicon vapor. The supersaturated niobium vapor starts
to generate stable nuclei which are the embryos of nanoparticles; and the neclei grow up to niobium-richnanoparticles (Figs.1(a) and 1(b)). After silicon vapor becomes supersaturated, the vapors of niobium andsilicon co-condense on the existing nanoparticles (Figs.1(c) and 1(d)), because the saturation pressure ofniobium is much lower than that of silicon. Following the consumption of niobium vapor, silicon vapor isconsumed and consequently the silicon content of the nanopowder becomes larger (Figs.1(e) and 1(f)).
During this process, coagulation among the nanoparticles also takes place simultaneously. As a result of
this growth behavior, the Nb-Si system produces a nanopowder having wide ranges of the size and thesilicon content because of the time lag of co-condensation between niobium and silicon. Figure 1 also tellsthat the majority of the nanoparticles have the diameters around 10 nm and the silicon content of 50 at.%.
It is noteworthy that the other smaller nanoparticles have smaller contents of silicon. These features are

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SENAC Vol. 47, No. 4（2014. 10）(a) 3200 K(b) 3100 K(c) 2700 K(d) 2500 K(e) 2300 K(f) 2100 K

Fig. 1 Growth process of Nb-Si nanopowder

Computational prediction of niobium-silicide nanopowder growthin thermal plasma synthesis— 15 —

determined by the balance of the particle size and the condensation rate which is a function of the particle
size as described in Eq. (2).

5. Summary
The high performance of supercomputing systems has made it feasible to clarify multi-scale physics. As
a recent application of supercomputing, a growth process of niobium-silicide nanopowder from theprecursory binary vapors of niobium and silicon in a thermal plasma environment has been computed
using a unique mathematical model. The collective and simultaneous growth behavior through nucleation,
co-condensation and coagulation has been clarified. Because of the time lag of binary co-condensation
between niobium and silicon, the synthesized nanopowder has wide ranges of the size and the silicon
content. The nanopowder also has some nanoparticles deviated from the mean size and the mean silicon
content.

Acknowledgements
The results of this work were obtained using the supercomputing resources of Cyberscience Center,
Tohoku University.
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