Radio Frequency Thermal Plasma: The Cutting Edge Technology in Production of Single-Walled Carbon Nanotubes (original) (raw)

Radio Frequency Thermal Plasma: The Cutting Edge Technology in Production of Single-Walled Carbon Nanotubes*

Sanaz ARABZADEH ESFARJANI**, Javad MOSTAGHIMI**, Keun Su KIM ***, Ali SHAHVERDI**** and Gervais SOUCY****
* Department of Mechanical and Industrial Engineering, University of Toronto 5 King’s College Rd., Toronto, M5S 3G8, Canada
E-mail: mostag@mie.utoronto.ca
*** Steacie Institute for Molecular Sciences, National Research Council 100 Sussex Dr., Ottawa, Ontario, K1A 0R6, Canada
**** Department of Chemical and Biotechnological Engineering, Université de Sherbrooke 2500, boul. de l’Université Sherbrooke, Sherbrooke, J1K 2R1, Canada

Abstract

The radio frequency (RF) inductively coupled plasma technique is a new and promising synthesis method of single-walled carbon nanotubes (SWCNTs) at large scales, for industrial and commercial applications. In this method, a mixture of carbon black and metal catalysts is directly vaporized by the RF plasma. Subsequently, inside the reactor chamber and under a controlled temperature gradient, carbon-metal clusters are formed and become the potential sites for nucleation and growth of SWCNTs. In this process, the local plasma properties and the thermo-fluid field in the system affect the yield rate of SWCNTs, and therefore it is important to find an appropriate operating condition, which maximizes the yield rate. Numerical modeling in conjunction with experimental studies can help investigate the contribution of the thermo-fluid field and process parameters to the formation of catalyst nanoparticles and carbon nanotubes in the induction thermal plasma system. The goal of this research is to carry numerical study of SWCNT growth in a RF induction thermal plasma system with a suitable chemistry model. This model is employed to investigate the influence of the thermo-fluid field and gas-phase reactions on carbon nanotube growth and to predict the SWCNT yield rate as a function of operating conditions.

Key words: Single-Walled Carbon Nanotubes (SWCNTs), Radio Frequency (RF) Induction Thermal Plasma, Numerical Modeling

1. Introduction

One of the revolutionary discoveries in nanoscience and nanotechnology is single-walled carbon nanotubes (SWCNTs) (1){ }^{(1)}. The unique electronic, mechanical, optical and chemical properties of this new generation of nanomaterials have opened opportunities in many engineering and medical applications. However, the costly manufacturing process of high quality SWCNTs has hindered the practical applications. As the industrial demand for SWCNTs continues to grow, it becomes crucial to develop economically efficient synthesis methods to produce SWCNTs of high quality at low cost. Several laboratory-scale methods have been proposed to produce SWCNTs. Examples

include arc discharge and chemical vapor decomposition (2){ }^{(2)}, laser ablation (3){ }^{(3)}, and different thermal plasma techniques (4){ }^{(4)}.

Most recently, the radio-frequency (RF) induction thermal plasma has shown distinctive advantages over conventional methods, making this technology a promising breakthrough for advancing the industrial fabrication of SWCNTs. Radio Frequency (RF) induction thermal plasmas operated by inductively heating the plasma. Electrical energy is provided to the plasma by passing an RF current through a copper coil wound around a dielectric tube, which is usually ceramic. As the electro-magnetic fields penetrate through the plasma, they lose electrical energy through joule heating. The excitation frequency is typically between 200 kHz and 40 MHz and the power levels at laboratory units vary between 30−50 kW(5)30-50 \mathrm{~kW}^{(5)}. The electrode-less operation of RF plasma torch enables the system to operate continuously and to generate final products clean from electrode erosion. Wang et al. (6){ }^{(6)} used an 1.67MHz−30 kW1.67 \mathrm{MHz}-30 \mathrm{~kW} RF inductively coupled plasma jet at the pressure range of 10−20kPa10-20 \mathrm{kPa} to primarily produce fullerenes by means of the direct evaporation of carbon powders. Watanabe et al. (7){ }^{(7)} experimentally studied the effects of metal addition on the formation of nanotube in RF thermal plasmas and then summarized the effectiveness of several metal catalysts (i.e. Ni,Co,Fe,La\mathrm{Ni}, \mathrm{Co}, \mathrm{Fe}, \mathrm{La} and Mo ) on carbon nanotube synthesis. Cota-Sanchez et al. (8){ }^{(8)} extensively explored a fullerene synthesis process by an RF plasma jet ( 2−5MHz,20−40 kW,40−66kPa2-5 \mathrm{MHz}, 20-40 \mathrm{~kW}, 40-66 \mathrm{kPa} ) to investigate the effects of the operating parameters and the type of raw materials employed on the formation of fullerene and other types of carbon nanostructures. Keun Su et al. (9){ }^{(9)} used an experimental setup at University de Sherbrooke to produce SWCNTs by adding metallic particles to the carbon black powder carbon source. They demonstrated that carbon soot product which contains approximately 40wt%40 \mathrm{wt} \% of SWCNT can be continuously synthesized at the high production rate of 100 g/h100 \mathrm{~g} / \mathrm{h}, while the quality of the tubes were comparable to those from laser ablation methods.

The high energy density and long residence time of the thermal plasma favor complete evaporation of the solid precursors, which in turn influences the quality and quantity of the end product. The direct evaporation of carbon black and metal catalysts mixtures is obtained at the exit of the plasma torch. Subsequently, inside the reactor chamber under a controlled temperature gradient environment, of vapors results in the formation and growth of catalyst nanoparticle aggregates and bundles of SWCNTs on the aggregates. Studies have shown that the size distribution and chemical composition of the catalysts nanoparticles, as well as the plasma and thermo-fluid fields in the system, affect the nanotubes diameter, growth rate, morphology and microstructure (10,11){ }^{(10,11)}. The optimization procedure of the RF plasma system requires a more complete understanding of the SWCNT formation mechanism and its relationship with the local plasma properties and thermo-fluid field in the system (10){ }^{(10)}. Numerical modeling in conjunction with experimental studies can help provide a fundamental understanding of the process parameters in order to optimize the system.

The present research seeks to further develop numerical modeling of nucleation and growth of metal nanoparticle and SWCNTs, so that models can better approximate industrially-relevant operating conditions and thus provide insights for the optimal and efficient fabrication of nanoparticle and SWCNTs in RF plasma systems. In this paper, we present the results of a 2D axisymmetric numerical model of an RF induction plasma torch and the reactor to investigate the distributions of the temperature, and species concentration fields established in the RF plasma torch and the reactor. The SWCNT chemical kinetic mechanism proposed by Scott (12){ }^{(12)} is also analyzed through zero-dimensional perfectly stirred reactor (PSR) simulations using the inlet boundary information from our thermo-flow field simulations. The formation of several intermediate species such as C2\mathrm{C}_{2} and C3\mathrm{C}_{3} is observed in the simulations. In the future work, we will incorporate the SWCNT chemistry model into the 2-D thermo-fluid model to

obtain more realistic results. The goal is to demonstrate the challenges in the modeling of the entire process and to present the ongoing research on the experimental and numerical parts.

Nomenclature

Subscripts

2. Experimental setup

As shown in Fig. 1, the synthesis system mainly consists of a plasma torch, a reaction chamber and a filtration system. The torch (PL-50, TEKNA Plasma Systems, Sherbrooke, Quebec, Canada) is driven by a 60 kW RF power supply (Lepel Co.) operated at an oscillator frequency of 3 MHz . The geometry and dimensions of the torch used in this work can be found in the Appendix. Three different gas streams of powder carrier, central, and sheath gases are introduced into the induction plasma torch. The central gas is injected with a swirl component to stabilize the plasma, whereas the sheath and powder carrying gases are injected without swirl. The powder carrying gas allows for efficient injection of mixtures of carbon black and metallic catalyst powder (nickel). The reaction chamber includes a graphite liner, and a thermal insulator, which are employed for an active control over maximum achievable temperature, background temperature, and cooling rate. For the completion of the SWCNT growth, the reaction zone is followed by a quench system in which the reaction gases are cooled down through heat exchange with the water-cooled reactor wall. The final products are then collected in a filtration chamber, which consists of three porous metallic filters. A typical operating condition employed for this numerical study is summarized in Table 1.

Fig. 1 Schematic of RF plasma system for the continuous production of SWCNTs at large scales (10){ }^{(10)}.

Table 1 A typical operating condition employed in this numerical study

3. Model Description

A complete mathematical model of the problem includes the prediction of the flow field and temperature field in the plasma, prediction of the particle trajectories and their temperature history, and the thermo-flow field in the reactor.

3.1. Radio Frequency Plasma Model

A 2-dimensional, axi-symmetric model for simulations of plasma generation by an RF induction plasma torch has been developed based on the procedure described by Mostaghimi et al. (13){ }^{(13)} and Proulx et al. (14){ }^{(14)}. The following assumptions were considered in modeling the induction plasma: (i) two-dimensional, axisymmetric and steady-state flow; (ii) quasi-neutral and optically thin plasma in a local thermodynamic equilibrium (LTE) state; (iii) negligible displacement current and viscous dissipation; (iv) planar coils; (v) laminar flow; (vi) consideration of the momentum and energy source terms due to the particle loading. Based on these assumptions, the plasma equations are described by steady-state conservation equations for the transport of mass, momentum, energy, and concentration of species coupled with electromagnetic field equations as follows:

Conservation of Mass

1ρDρDt+∇⋅u=0\frac{1}{\rho} \frac{D \rho}{D t}+\nabla \cdot \boldsymbol{u}=0

Conservation of Momentum

ρDuDt=−∂p∂z+∇⋅μ∇u+1r∂∂r(μr∂v∂r)+(μ∂u∂z)+(j×B)zρDvDt=−∂p∂r+∇⋅μ∇v+1r∂∂r(μr∂v∂r)+(μ∂u∂z)−2μvr2+ρw2r+(j×B)zρDwDt=∇⋅μ∇w−1r∂∂r(μw)+(μ∂∂z(wr))−ρvwr\begin{aligned} & \rho \frac{D u}{D t}=-\frac{\partial p}{\partial z}+\nabla \cdot \mu \nabla u+\frac{1}{r} \frac{\partial}{\partial r}\left(\mu r \frac{\partial v}{\partial r}\right)+\left(\mu \frac{\partial u}{\partial z}\right)+(\mathbf{j} \times \mathbf{B})_{z} \\ & \rho \frac{D v}{D t}=-\frac{\partial p}{\partial r}+\nabla \cdot \mu \nabla v+\frac{1}{r} \frac{\partial}{\partial r}\left(\mu r \frac{\partial v}{\partial r}\right)+\left(\mu \frac{\partial u}{\partial z}\right)-\frac{2 \mu v}{r^{2}}+\frac{\rho w^{2}}{r}+(\mathbf{j} \times \mathbf{B})_{z} \\ & \rho \frac{D w}{D t}=\nabla \cdot \mu \nabla w-\frac{1}{r} \frac{\partial}{\partial r}(\mu w)+\left(\mu \frac{\partial}{\partial z}\left(\frac{w}{r}\right)\right)-\frac{\rho v w}{r} \end{aligned}

Conservation of Energy

ρDhDt=∇⋅κ∇T+j⋅E−R˙\rho \frac{D h}{D t}=\nabla \cdot \kappa \nabla T+\mathbf{j} \cdot \mathbf{E}-\dot{R}

Species Concentration Equation

ρDYeDt=∇⋅(ρDe)∇Ye+S2\rho \frac{D Y_{e}}{D t}=\nabla \cdot\left(\rho D_{e}\right) \nabla Y_{e}+S^{2}

For the Lorentz force and joule heating terms appearing in the momentum and energy conservation equations, it is necessary to calculate the electromagnetic field distributions induced by the alternating coil current. The electromagnetic fields and current density are determined by using the vector potential form of Maxwell’s equations. The magnetic induction, B\boldsymbol{B}, and the vector potential, A(r,t)\mathbf{A}(\mathbf{r}, t), are defined as,

B=μ0H≡∇×A(r,t)\boldsymbol{B}=\mu_{0} \boldsymbol{H} \equiv \nabla \times \boldsymbol{A}(\boldsymbol{r}, t)

where H\boldsymbol{H} is the magnetic field intensity and r\boldsymbol{r} is the position vector. Since the applied current is assumed to be sinusoidal, it is reasonable to expect that the induced fields are also sinusoidal, i.e.,

A(r,t)=AC(r,t)eωt\mathbf{A}(\mathbf{r}, t)=\mathbf{A}_{C}(\mathbf{r}, t) e^{\omega t}

where ω\omega is the radial frequency of the fields, and AC\boldsymbol{A}_{C} is the complex amplitude of the vector potential. For a standard induction plasma torch with conventional coil geometry, it is reasonable to assume that the electric field and the vector potential have only a tangential component. The electric field (Eθ)\left(E_{\theta}\right), axial, and radial components of the magnetic field ( HzH_{z} and HrH_{r}, respectively) and vector potential (Aθ)\left(A_{\theta}\right) are correlated as follows:

Eθ=−iωAθμeHz=1r∂∂r(rAθ)μaHr=∂Aθ∂z\begin{aligned} & E_{\theta}=-i \omega A_{\theta} \\ & \mu_{e} H_{z}=\frac{1}{r} \frac{\partial}{\partial r}\left(r A_{\theta}\right) \\ & \mu_{a} H_{r}=\frac{\partial A_{\theta}}{\partial z} \end{aligned}

The Joule heating and axial and radial Lorentzian forces are written as:

P˙=12σ[EθEθ∗]Fz=12μ0σR[EθHr∗]Fy=12μ0σR[EθHr∗]\begin{aligned} & \dot{P}=\frac{1}{2} \sigma\left[E_{\theta} E_{\theta}^{*}\right] \\ & F_{z}=\frac{1}{2} \mu_{0} \sigma \mathfrak{R}\left[E_{\theta} H_{r}^{*}\right] \\ & F_{y}=\frac{1}{2} \mu_{0} \sigma \mathfrak{R}\left[E_{\theta} H_{r}^{*}\right] \end{aligned}

In the above equations, the superscript * represents the complex conjugate.

3.2. Plasma-Injected Particle Interaction Model

In the induction thermal plasma process, the carbon precursor material is directly evaporated in the plasma plume, and therefore, the yield rate of SWCNT in the reactor can be affected by the evaporation efficiency of the injected particles. In a typical plasma jet reactor used in thermal plasma processing, the only major force acting on the injected particles in the plasma is the viscous drag force. The local surroundings of injected particles are changing rapidly in the thermal plasma processing. For this reason the Besset history term can also be considered in the equation of motion to account for the particle motion under strongly non-steady conditions. It has been found that for particles smaller than 100 microns this effect is negligible compared to the viscous effect. However, under low relative velocity conditions, during the final stages of processing, the influence becomes more important. The momentum exchange between particles and plasma is calculated by Newton’s law:

mpdupdt=π8dp2ρCD(u−up)∣u−up∣m_{p} \frac{d \boldsymbol{u}_{p}}{d t}=\frac{\pi}{8} d_{p}^{2} \rho C_{D}\left(\boldsymbol{u}-\boldsymbol{u}_{p}\right)\left|\boldsymbol{u}-\boldsymbol{u}_{p}\right|

where CD , the drag force coefficient (15){ }^{(15)}, is calculated by,

CD=248π;61+8π+0.4)(ρcμeρaμa)0.4t[11+(2−aa)(ra1+ra)APrpKn]0.45(4.7−4.33pr)0.2tC_{D}=\frac{24}{8 \pi} ; \frac{6}{1+\sqrt{8 \pi}}+0.4)\left(\frac{\rho_{c} \mu_{e}}{\rho_{a} \mu_{a}}\right)^{0.4 t}\left[\frac{1}{1+\left(\frac{2-a}{a}\right)\left(\frac{r_{a}}{1+r_{a}}\right) \frac{A}{\mathrm{Pr}_{p}} K n}\right]^{0.45}(4.7-4.33 \mathrm{pr})^{0.2 t}

where Re is the Reynolds number based on the diameter of particles. The second, third and fourth terms are the correction coefficients for considering the temperature variation in the boundary layer, the non-continuum effects (Knudson effects), and the effect of

particle non-sphericity, respectively. Sphericity, ψ′′\psi^{\prime \prime}, is defined as the ratio of surface area of a sphere having the same volume to the surface area of the particle. In thermal plasma processing, heat and mass transfer between the plasma and the injected particles is the most critical mechanism. The particles that are injected into the plasma are first heated; some are partially melted and leave the plasma, and some will be evaporated. The rest will be melted or cooled down to the solid state again. Heat and mass exchanges between plasma and particles are defined as follows:
Q=πdp2hr(T−Tp)−πdp2εpσsb(Tp4−Te4)Q=\pi d_{p}^{2} h_{r}\left(T-T_{p}\right)-\pi d_{p}^{2} \varepsilon_{p} \sigma_{s b}\left(T_{p}^{4}-T_{e}^{4}\right)
where the heat transfer coefficient, hrh_{r}, is given with the Nusselt number (15){ }^{(15)} :
Nu=(2+0.6Re⁡T1/2Pr⁡T1/2)(ρoμoρpμp)0.6(cpsbcpp)0.381ψN u=\left(2+0.6 \operatorname{Re}_{T}^{1 / 2} \operatorname{Pr}_{T}^{1 / 2}\right)\left(\frac{\rho_{o} \mu_{o}}{\rho_{p} \mu_{p}}\right)^{0.6}\left(\frac{c_{p_{s b}}}{c_{p_{p}}}\right)^{0.38} \frac{1}{\psi}
where ReTR e_{T} and PrTP r_{T} are Reynolds and Prandtl numbers based on the film temperature, respectively. The second and third terms are corrections for taking into account the temperature variation in the boundary layer. The boundary conditions of the RF plasma torch and the particle injection in the plasma can be found in the Appendix.

3.3. Thermo-Flow Field in the Reactor

The thermo-flow field equations, solved inside the reactor, were continuity, conservation of momentum, conservation of energy, and conservation of species. Diffusion coefficients of the gaseous mixture were calculated based on the mixture model in FLUENT (16){ }^{(16)}. The k - c\boldsymbol{c} turbulent model was implemented for the flow inside the reactor. The magnetic and electric fields in the reactor are insignificant and can be neglected in computations. To predict the growth of SWCNTs in this zone, equations for the SWCNT chemistry should be solved along with the thermo-flow field. For the time being, the chemical reactions are not considered in this preliminary study but we will include the chemistry model in the next step.

3.4. Single-walled Carbon Nanotube Growth Model

In the formation process of SWCNTs, a precursor of carbon vapor and vaporized metal catalysts are required. In this process, clusters of carbon and metal catalysts condense to form SWCNTs, as well as impurities such as soot and carbon coated metal clusters. Nucleation and growth of carbon nanotubes are not well understood and there are several possible scenarios for carbon and metal catalysts to condense in a reactor and then grow into nanotubes. There are different models based on molecular dynamics (17,18){ }^{(17,18)} or Monte Carlo (19){ }^{(19)} approaches, describing the nucleation and growth of SWCNTs from the metal nanoparticles. However, implementing those models into the plasma and particle models is quite challenging because of their large differences in physical scales.

In the previous work by Keun Su et al. (16){ }^{(16)}, a simplified model was adopted based on supersaturation of metal catalyst nanoparticles. In the model, the nucleation of SWCNTs is started when the droplet particles are supersaturated by carbon atoms. The surplus carbon atoms are assumed to be precipitated out of the surface of the droplets to form either SWCNTs or the graphene layers which eventually poison the catalyst. The metal catalyst nanoparticles generation process was described by aerosol general dynamic equation (GDEs). The GDE can be written as:
∂n(vd)∂t=−∇⋅(vn(vd))−∇⋅(uiqn(vd))+∇⋅(Dd(vd)∇n(vd))+I⋅δ(vd−vd∗)−∂[Gn(vd)]∂vd\frac{\partial n\left(v_{d}\right)}{\partial t}=-\nabla \cdot\left(\boldsymbol{v} n\left(v_{d}\right)\right)-\nabla \cdot\left(\boldsymbol{u}_{i q} n\left(v_{d}\right)\right)+\nabla \cdot\left(D_{d}\left(v_{d}\right) \nabla n\left(v_{d}\right)\right)+I \cdot \delta\left(v_{d}-v_{d}^{*}\right)-\frac{\partial\left[G n\left(v_{d}\right)\right]}{\partial v_{d}}

−∫0∞β(vd,vd′)n(vd)n(vd′)dvd′+12∫0∞β(vd′,vd−vd′)n(vd′)n(vd−vd′)dvd′-\int_{0}^{\infty} \beta\left(v_{d}, v_{d}^{\prime}\right) n\left(v_{d}\right) n\left(v_{d}^{\prime}\right) d v_{d}^{\prime}+\frac{1}{2} \int_{0}^{\infty} \beta\left(v_{d}^{\prime}, v_{d}-v_{d}^{\prime}\right) n\left(v_{d}^{\prime}\right) n\left(v_{d}-v_{d}^{\prime}\right) d v_{d}^{\prime}

where the first, second, and third terms describe mass transfer inside a control volume caused by convection, thermophoresis, and diffusion processes, respectively. The fourth term is the metal nuclei generation term through the homogeneous nucleation. The fifth term, represents the growth of carbon metal droplets by the co-condensation process. The sixth and seventh terms stand for the decrease and growth of carbon metal droplets through the coagulation process. In solving the GDEs, two methods of bin sectional and method of moments were used and the nucleation and growth of SWCNTs for each method was calculated. From the simulations, the temperature and mean droplet size profiles were calculated along the reactor axis. The mean diameters of the droplets were predicted to be around 10 nm , Fig. 2, at the exit of the reactor system which is in fairly good agreement with the TEM images of the SWCNT soot (10){ }^{(10)}.

Fig. 2 Axial profiles of the temperature and mean particle diameter calculated along the axis of the reactor system (10){ }^{(10)}.

In this model and several other schemes proposed for modeling the formation of carbon nanotube, the chemical kinetics of the reactions of species in the system were not considered. This is due to the complex mixture of species and the nature of conditions in the processing systems. The temperature in the system varies from several thousand degrees Kelvin down to near room temperature. Within these temperature ranges, hundreds of species can exist ranging from atomic carbon, to large clusters of carbonaceous soot, metallic catalyst atoms, metal clusters, and combinations of carbon-metal clusters. Most of the data for the chemical kinetics of the reactions and the thermodynamic properties of clusters are not easily available and therefore, have been approximated. All of these factors have made the computational simulations of the synthesis system very challenging.

In this work, the kinetic model proposed by Scott (12){ }^{(12)}, reproduced in Table 2, has been used. Based on the model, the nickel model (20){ }^{(20)} is added to the set of reactions of carbon, based on the fullerene model of Krestinin et al. (21){ }^{(21)}, to allow production of fullerene and other large carbon clusters. Carbon and nickel clusters combine to form the nuclei for the growth of carbon nanotubes. The growth rate of carbon/nickel clusters can be obtained by the aerosol theory of Rao et al. (22){ }^{(22)}. Carbon clusters in the form of fullerene (C60)\left(\mathrm{C}_{60}\right) and soot are assumed to be the precursors to form carbon/nickel clusters. These clusters are the

species that nucleate and grow carbon nanotubes. In the reduced version of the kinetic model, 16 gas species and 38 gas-phase reactions are considered. In the present work, the kinetic mechanism is implemented and tested for a range of temperatures of relevance to the thermal plasma processing system, in a zero-dimensional PSR system.

3.5. Computational Domain and Numerical Method

The computational domain in the present work consists of three zones; a RF plasma torch, a reactor and a quenching zone. The electromagnetic field equations are solved using a fourth-order Runge-Kutta method. The SIMPLER algorithm developed by Patankar (23){ }^{(23)} is employed to obtain solutions to the governing equations. The effect of particle shapes is considered in the last term. The so-called Particle-Source-In-Cell (PSI-Cell) method (24){ }^{(24)} is used to numerically solve the plasma-particle interaction equations. The plasma generation and particle tracking were solved by an in-house code (25){ }^{(25)}.

The gas flow (including argon, helium, carbon and nickel vapors) exits the plasma torch and enters the reactor. For the simulation inside the reactor, flow and concentration profiles at the exit of the plasma torch were extracted from the results calculated from the in-house code and used as the boundary conditions for FLUENT 12.1.1 (16){ }^{(16)}. The 2D axisymmetric thermo-flow fields inside the reactor were simulated using FLUENT 12.1.1 (16){ }^{(16)}. As mentioned above, we did not consider the SWCNT chemistry in this 2-D simulation but we have performed 0-D chemical kinetic simulations separately by using CHEMKIN (26){ }^{(26)} code. The current research efforts are devoted to generating a coupled flow-chemistry model.

Table 2 List of kinetic reaction rates, reproduced from (12){ }^{(12)}.
(k=AT∗∗bexp⁡[−E/RT])\left(\mathbf{k}=\mathbf{A} \mathbf{T}^{* *} \mathbf{b} \exp [-E / R T]\right)

4. Results and Discussion

Figure 3 shows the computed distributions of the temperature inside the RF plasma torch. The off-axis high-temperature zone inside the torch is mainly due to skin depth effect. The injected particles inside the RF torch create a lower temperature region along the centerline close to the injection probe. The presence of carbon and nickel vapors in the discharge zone of the plasma cause the high temperature zone (i.e., T>7000 K\mathrm{T}>7000 \mathrm{~K} ) to subside close to the injection probe.

Fig. 3 Temperature contours of the plasma gas

In the reaction chamber, Fig. 4, the plasma jet expands and the off-axis high-temperature zone disappears as a result of diffusion. The temperature field remains around 4000 K or above over the reaction zone. At the end of the graphite liners (Z=0.5(\mathrm{Z}=0.5 m)\mathrm{m}) and in the quenching zone, the temperature decreases rapidly to around 1000 K due to the heat exchange with the water-cooled reactor walls. The plasma gas composition plays an important role in the distributions of temperature and species inside the plasma torch and reactor. It has been shown that the mixture of Ar-He plasma generates a higher temperature gradient compared to the pure Ar-plasma (9){ }^{(9)}. This is due to the higher thermal conductivity of He compared to Ar. Fig. 5 shows the mass fraction distributions of the carbon and nickel vapors inside the reactor as the vapor mixture is convected and diffuses

throughout the reactor. The concentration decrease calculated based on diffusion and convection mechanisms is slightly less than the real condition in which chemical reactions participate in consuming the species in the reactor. Adding chemical reactions to the model is part of ongoing research efforts.

Fig. 4 Temperature contours in the reactor

Fig. 5 C/Nickel mass fraction by diffusion and convection mechanisms

The axial profiles of species vapor concentrations and temperature along the centerline are presented in Fig. 6. The mass fraction profiles of the carbon and nickel vapors reach their maximum values at the entrance of the reaction zone and then decrease and remain constant along the reactor axis. The temperature rises to about 5400 K close to the inlet of the reactor from the flow coming from the RF plasma torch and then starts to decrease upon entering the quenching zone in the reactor.

Fig. 6 Axial Temperature, C/Nickel mass fraction profiles in the reactor

The carbon and nickel vapors produced in the hot regions are then transported to the cooler regions via diffusion and convection, and the carbon and nickel atoms start to form various carbon clusters, nickel clusters, nickel-carbon composite clusters, and SWCNTs. In the RF plasma synthesis system, the production rate of SWCNTs is strongly affected by the temperature profiles in the RF plasma torch and in the reactor because temperature effects govern the formation and destruction of several intermediate and stable chemical species in the system.

In order to examine the effect of the temperature gradient on the chemistry model of SWCNT formation, zero-dimensional PSR simulations were conducted with the chemical mechanism in Table 2 by CHEMKIN (26){ }^{(26)}. The chemistry model was computed using boundary conditions derived from the inlet of the reactor. In our PSR simulations, the reactor was assumed to be continuously fed by a mass flow of 1.2742 g/s.1.2742 \mathrm{~g} / \mathrm{s} .. The composition of the inlet flow is considered with a ratio of Ar/He/C/Ni\mathrm{Ar} / \mathrm{He} / \mathrm{C} / \mathrm{Ni} of: 76.4at%/22.076.4 \mathrm{at} \% / 22.0 at%/1.5at%/0.1at%\mathrm{at} \% / 1.5 \mathrm{at} \% / 0.1 \mathrm{at} \% and is based on the plasma code, . The initial composition of species in the PSR was imposed as Ar/He:25at%/75at%\mathrm{Ar} / \mathrm{He}: 25 \mathrm{at} \% / 75 \mathrm{at} \%. The temperature inside the PSR is considered uniform and the calculations were carried out for several cases of different reactor temperatures ranging from 4,500 K4,500 \mathrm{~K} down to 1,000 K1,000 \mathrm{~K}. Fig. 7 shows the preliminary results of mole fraction versus temperature of two important intermediate species in the formation and growth of carbon nanotubes, i.e., C2\mathrm{C}_{2} and C3\mathrm{C}_{3}. These species start to appear around temperatures of 2,500−4,000 K2,500-4,000 \mathrm{~K} with a maximum peak around 3,500 K and as the temperature inside the reactor drops below 2,000 K2,000 \mathrm{~K}, the concentration of these carbon species drop significantly. The purpose of these computations is to test the stability and suitability of the chemical mechanism at the temperatures and compositions relevant to the reactor system. Although the mechanism was found to be computationally stable, reactions coefficients should be adjusted and validated at higher temperatures, which occur in the reactor system.

Fig. 7 Mole fractions of C2\mathrm{C}_{2} and C3\mathrm{C}_{3} species in PSR simulations

5. Summary

We have performed CFD simulations of the SWCNT growth in the RF induction thermal plasma process. The 2D CFD simulations show that the thermo-fluid fields inside the torch and the reaction chamber have steep gradients in both radial and axial directions. The strong dependency of the SWCNT yield on the process temperature has also been confirmed by the kinetic chemistry simulations. Current work seeks to incorporate kinetic

reactions into our simulations in order to obtain more detailed information from the CFD results, and model the coupled effects of chemistry and fluid flow.

Acknowledgments

This work was supported by the Natural Science and Engineering Research Council (NSERC) of Canada and the Ontario Ministry of Research and Innovation. The authors wish to thank Dr. Carl D. Scott and Dr. Samir Farhat for supplying the CHEMKIN chemical kinetic files.

Appendix

1. Cross-sectional view of an RF induction thermal plasma torch (PL-50, TEKNA) employed for the continuous synthesis of SWCNTs.

2. RF Plasma Torch Boundary Conditions

Inlet boundary condition (z=0)(z=0) :

u={Q1/πr12r<r10r1≤r≤r2Q2/π(r22−r22)r2<r≤r1Q3/π(R02−r32)r3<r≤R0u=\left\{\begin{array}{cc} Q_{1} / \pi r_{1}^{2} & r<r_{1} \\ 0 & r_{1} \leq r \leq r_{2} \\ Q_{2} / \pi\left(r_{2}^{2}-r_{2}^{2}\right) & r_{2}<r \leq r_{1} \\ Q_{3} / \pi\left(R_{0}^{2}-r_{3}^{2}\right) & r_{3}<r \leq R_{0} \end{array}\right.

Centerline (r=0)(r=0) :

∂u∂r=v=∂h∂r=AR=AI=0\frac{\partial u}{\partial r}=v=\frac{\partial h}{\partial r}=A_{R}=A_{I}=0
Walls (r=R0):‾\underline{\text { Walls }(\boldsymbol{r}=R_{0}):}

AR=μ0I2πRcR0∑i=1nG(ki)+μ0ω2π∑i=1cvriR0σiAl,iSiG(ki)AI=−μ0ω2π∑i=1cvriR0σiAR,iSiG(ki)\begin{aligned} & A_{R}=\frac{\mu_{0} I}{2 \pi} \sqrt{\frac{R_{c}}{R_{0}}} \sum_{i=1}^{n} G\left(k_{i}\right)+\frac{\mu_{0} \omega}{2 \pi} \sum_{i=1}^{c v} \sqrt{\frac{r_{i}}{R_{0}}} \sigma_{i} A_{l, i} S_{i} G\left(k_{i}\right) \\ & A_{I}=-\frac{\mu_{0} \omega}{2 \pi} \sum_{i=1}^{c v} \sqrt{\frac{r_{i}}{R_{0}}} \sigma_{i} A_{R, i} S_{i} G\left(k_{i}\right) \end{aligned}

u=v=0λ∂T∂r=λcδw(Tc−Tw)\begin{aligned} & u=v=0 \\ & \lambda \frac{\partial T}{\partial r}=\frac{\lambda_{c}}{\delta_{w}}\left(T_{c}-T_{w}\right) \end{aligned}

3. Particle injection Boundary Conditions

Convection and radiation boundary conditions:

Kp∂T∂r∣r=t0=h(T∞−T∞)−εσT∞4K_{p}\left.\frac{\partial T}{\partial r}\right|_{r=t_{0}}=h\left(T_{\infty}-T_{\infty}\right) \quad-\varepsilon \sigma \quad T_{\infty}^{4}

where r0r_{0} is the outer radius of the particle.
Interface between two phases:

Kp∂T∂r∣r=ti′′=Kp∂T∂r∣r=ti′′+ρLfdridtK_{p}\left.\frac{\partial T}{\partial r}\right|_{r=t_{i}^{\prime \prime}}=K_{p}\left.\frac{\partial T}{\partial r}\right|_{r=t_{i}^{\prime \prime}}+\rho L_{f} \frac{d r_{i}}{d t}

where rir_{i} is the radius of the particle at the interface at which T=Tmeeting T=T_{\text {meeting }}.
Diffusion boundary of vaporizing particle ( T<Tboiling T<T_{\text {boiling }} ):

Kp∂T∂r∣r=t0=h(T∞−T∞)−εσT∞4−ρhmMLeLn(pp−pv)K_{p}\left.\frac{\partial T}{\partial r}\right|_{r=t_{0}}=h\left(T_{\infty}-T_{\infty}\right) \quad-\varepsilon \sigma T_{\infty}^{4}-\rho h_{m} M L_{e} L n\left(\frac{p}{p-p_{v}}\right)

Intense evaporation boundary:

Kp∂T∂r∣r=t0=h(T∞−T∞)−εσTw4+ρLedr0dtK_{p}\left.\frac{\partial T}{\partial r}\right|_{r=t_{0}}=h\left(T_{\infty}-T_{\infty}\right) \quad-\varepsilon \sigma T_{w}^{4}+\rho L_{e} \frac{d r_{0}}{d t}

where Tw=Tboiling T_{w}=T_{\text {boiling }}
Symmetry condition:

∂T∂r∣r=0=0\left.\frac{\partial T}{\partial r}\right|_{r=0}=0

Initial Condition:

Ti=0=Tcorrer gas T_{i=0}=T_{\text {correr gas }}

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