* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Three–dimensional Modelling of dc Arc Discharges for Carbon Nanostructure Production E. Tam and A. B. Murphy CSIRO Material Science and Engineering, P.O. Box 218, Lindfield NSW 2070, Australia Abstract: Discharges in helium between carbon electrodes at atmospheric or sub– atmospheric pressure have proved to be excellent sources of carbon nanostructures, such as graphene and carbon nanotubes. However, the formation mechanisms are poorly understood. A three–dimensional fluid mode of a dc arc discharge in helium that includes the carbon electrodes and their vaporization self–consistently has been developed; preliminary results are presented in this paper. This model allows the determination of the concentration of carbon species at all points in the arc. Results obtained by including and neglecting electrode vaporization are compared. Keywords: arc discharge, carbon nanotubes, graphene, modeling, computational fluid dynamics 1. Introduction 2. Model Carbon nanostructures, such as carbon nanotubes and graphene nanoribbons, have unique properties that have motivated many researchers to attempt their integration into advanced new devices. Potential applications include, but are not limited to, drug and gene delivery, hydrogen storage and electron field emission[1,2]. Arc discharges generally produce large volumes of high quality nanostructures (The ratio of intensity G–band peaks over the D–band peaks are higher when carbon nanotubes produced by arcs are examined using Raman spectroscopy), because the manufacturing process occurs at very high temperatures when compared to other methods [3,4]. However, even though arc discharges are already used to produce nanostructures commercially for scientific purposes, there are still challenges that must be overcome for applications outside the laboratory. These challenges include controlling the environment to minimize the large temperature gradients typically seen in an arc discharge, optimizing the energy inputs so there the arc produces just enough heat to ablate the electrodes and nucleate the nanostructures and, as with all other fabrication methods, controlling the growth of the desired nanostructure (such as diameter and chirality of single–walled carbon nanotubes). Figure 1. Schematic of the system modeled. A schematic of the system modeled in these simulations can be seen in Fig. 1. The computation domain is a rectangular prism. Room temperature helium is pumped into the system at a constant rate through the hollow graphite anode. The graphite cathode is roughly twice the diameter of the anode. The plasma is assumed to be incompressible and can be approximated with local thermal equilibrium (LTE). The heat transfer boundary conditions given by Tanaka and Lowke for a thermionic cathode and an anode were used . In addition, the cooling effects associated with the latent heat of vaporization were taken into account. The standard conservation and continuity equations, adapted from those of conventional computational fluid dynamics to take into account thermal plasma phenomena, are used. These include the mass continuity equation, charge continuity, the Navier– Stokes equation and energy conservation [6-11]. √ (5) In this binary gas system, the diffusive mass flux of carbon, relative to the mass–average velocity in a helium plasma, was determined using the combined diffusion coefficient method [9-11]: ( ) An equation for conservation of the carbon vapor mass fraction is also required. This is  ( ) (1) where is the mass density of the plasma, is the plasma velocity, is the mass fraction of carbon in the plasma, is mass flux of the carbon vapor, is the source term for the ablated carbon vapor and is the time. The carbon mass source term used in Eqn (1) is approximated to be  ( ) (2) at the plasma–electrode interface and 0 elsewhere. The variables , and are the mass of a carbon atom, the vaporization (or ablation) flux and the deposition flux respectively. The Hertz–Knudsen relation  √ (3) is used to determine the vaporization flux, where is the saturation vapor pressure of carbon, determined by Clausius–Clapeyron relation (4) where is the latent heat of the graphite and is the specific gas constant for carbon ⁄ ( ). The deposition flux is calculated using  (6) The finite difference method described by Patankar is used to numerically solve these equations in three dimensions . These equations are solved self consistently and the electrodes are included in the computational domain. The thermophysical properties of the helium–carbon mixtures were determined by Murphy [10,11]. Net radiative emission coefficients were determined using the methods of Cram, assuming a 1 mm absorbing region . 3. Results and Discussion Figure 2 shows the velocity streamlines and the current density in a system with and without carbon ablating from the electrodes. When carbon ablation is not considered, eddies form in the plasma around the electrodes, centered close to a local maximum of the current density. As there is no ablation or deposition occurring, the plasma flow must travel parallel to the solid surfaces (i.e. both electrodes and the chamber walls) in their immediate proximity. The two main driving forces of the plasma flow in this system are the magnetic pinch force and the influx of helium through the hollow anode. This influx determines the overall flow of the plasma. The magnetic pinch force, on the other hand, accelerates the plasma very strongly in the localized regions in which the current density is high. The magnetic pinch force contributes little to the overall trend of the plasma flow entering through the anode and leaving at the outlet; rather it is the cause of the eddies that form at the electrodes and force the plasma to move at very high local speeds. significantly increases the electrical conductivity in the cooler regions of the plasma, spreading the current over a larger region. Figure 3: The temperature distrbution of a dc arc discharge in He gas with graphite electrodes excluding carbon vapor evaporation from the electrodes. Scale bar is in kelvin. Figure 2: Current density and plasma stream lines of (a) arc with no carbon ablating from the electrodes and (b) arc with carbon ablating and depositing on to the electrodes. The colors represent current density on a logarithmic scale, with scale bar to the left of each figure (units in A/cm2). Note the discontinuities in current density and the kinks in the streamline that appear in (b) are artifacts due to the low number of iteration used in this particular run. When vaporization and deposition of carbon vapor is included, we can see some significant changes in the plasma flow. Whereas the plasma flow was parallel to the electrodes in their immediate vicinity, it is now perpendicular to the surface in regions where the electrode is hot. This shows that the large volume of vaporized carbon disrupts the flow close to the electrodes. In addition, the eddies that had formed when no carbon was being ablated are reduced in size when vaporization is taken into account. This is most likely due to the fact that the current density is not as high when carbon is ablated. The addition of carbon vapors to the plasma The temperature distribution in the plasma with no carbon vaporization is shown in Figure 3. When no carbon vaporization is considered, a global temperature maximum can be seen adjacent to the anode. The large plasma velocities and the eddies that form under these conditions increases the effects of convective cooling, creating very large temperature gradients in the immediate vicinity of the electrodes. While final results have not yet been obtained for the case in which the vaporization of the electrodes is included, it is expected that the carbon vapor that is introduced into the plasma will cool the arc region, as carbon has significantly higher radiative emission and higher electrical conductivity (leading to lower current densities). The size reduction of the eddies means there is little convective transport of heat, leading to a reduction in the temperature gradients. Figure 4 shows the mass fraction of carbon due to vaporization. Clearly the carbon concentration is greatest in the regions immediately below the anode. Inclusion of diffusion in the calculation is expected to smear out the minimum in carbon mass fraction on the arc axis. The lower maximum temperature (which is still larger than carbon’s sublimation point), the smaller temperature gradients and the reduction in convective motion of the plasma are the key to the growth of high quality nanostructures. Nanostructures of different morphologies all nucleate and grow at specific temperatures. In the case of carbon, no nanostructure will nucleate at temperature greater than ~3915 K. Without the presence of catalysts, the temperatures at which various useful nanostructures can be nucleated are in the range from around 1500 to 2500 K, after which a narrow temperature range is optimal for their growth [15,16]. Assuming that the nanoparticles that form in the arc travel with plasma flow, ideally the maximum arc temperature should be just above optimum nucleation temperatures for the required nanostructures, and the temperature of the plasma through which the nanoparticles travel as they grow should stay fairly close to the optimum growth temperature of the desired nanostructure. This will ensure high the production of high volumes of quality nanostructures. the plasma from vaporization of the anode. The decrease in the eddy size weakens the convective cooling, which will reduce the temperature gradients. This improves the suitability of such arcs for the growth of carbon nanostructures. However, there is a large amount of further work require to tailor the arc conditions so that selective growth of nanostructures can be achieved. References  D. Cai et al. Nat. Methods 2, 449–454 (2005).  A. Sidorenko, T. Krupenkin, A. Taylor, P. Fratzl, and J. Aizenberg. Science 315, 487–490 (2007).  Y. Ando, X. Zhao, K. Hirahara, K. Suenaga, S. Bandow, and S. Iijima. Chem. Phys. Lett. 323, 580 – 585 (2000).  J. Qiu, Y. Li, Y. Wang, T. Wang, Z. Zhao, Y. Zhou, F. Li, and H. Cheng. Carbon 41, 2170 – 2173 (2003).  M. Tanaka and J. J. Lowke. J. Phys. D 40, R1 (2007).  J. J. Lowke, R. Morrow, and J. Haidar. J. Phys. D 30, 2033–2042 (1997).  A. B. Murphy. J. Phys. D 31, 3383–3390 (1998).  A. B. Murphy. Phys. Rev. Lett. 89, 025002 (2002). Figure 4: Mass fraction of He in the plasma when carbon ablation from the electrodes is included. Note that these results were obtained with only convective mixing taken into account (i.e. neglecting diffusion). 4. Conclusion The effect of the inclusion of carbon mass conservation on a dc arc discharge has been investigated. Comparisons to results obtained for a pure helium plasma with no carbon vaporization show differences in the plasma flow, current densities and temperature distribution of the plasma. One of the strongest differences between the two systems is the size reduction of the eddies when ablation is included. This is a consequence of the reduction of the current densities near the electrodes and of the relatively large volume of carbon entering  A. B. Murphy. Plasma Chem. Plasma Process. 20, 279–297 (2000).  A. B. Murphy. J. Phys. D 43, 434001 (2010).  A. Murphy. IEEE T. Plasma Sci. 25, 809 –814 (1997).  Y. Tanaka. In Conference Proceedings GD2010 (2010).  S. V. Patankar. Numerical Heat Transfer and Fluid Flow. Hemisphere Publishing Corporation (1980).  L. E. Cram. J. Phys. D 18, 401 (1985).  M. Keidar. J. Phys. D 40, 2388–2393 (2007).  K. Kim, A. Moradian, J. Mostaghimi, and G. Soucy. Plasma Chem. Plasma Process. 30, 91– 110 (2010).