Chapter 2. Arc discharge Fullerene Formation Mechanism
2.2 An arc-jet model to simulate the carbon cluster evolution during fullerene formation
In this section, we first present a simple model proposed by Dr. Lowell Lamb to simulate the arc-jet fullerene formation procedure (Fig. 2.1a). The model recognizes the whole carbon cluster evolution to happen in a jet of carbon vapor mixed with helium or other buffer gases (Fig. 2.1b). The jet starts from the erosion of the electrodes (made of graphite) with intense current, and the initial carbon vapor generated will be at a very high temperature ~ 5500 K, consisting only very small carbon fragments like C1, C2, and C3. As arc physics is actually a total different field, only simple description of this part will be provided in the form of parameters, like initial temperature, initial small carbon cluster concentrations, etc. Within the jet, complicated thermodynamics or turbulence will also be ignored. Instead, simple linear models for velocity, temperature, and pressure, as well as a cone geometry of the jet, will be adopted. Only collisions are considered happening during the cluster evolution (which will be modified in the later version), which requires a model for the cross section and diffusion rate for the clusters that collide with each other. One very important assumption of this model is that all the collisions will result in coalescence of the clusters. This assumption greatly simplifies the model but comes with severe consequences which would be replaced with more advanced and reasonable assumptions.
To program the jet model, every step we consider the things happen in a very thin slice shown in Fig. 2.1b. The slice will evolve from the small slot between the electrodes to the end of the jet, usually of a 3~10 cm distance, which depends on the pressure of the working chamber. The evolution of the slice is just the evolution of the carbon clusters until the formation of fullerenes.
During the evolution, the temperature, pressure, velocity of the clusters, and radius of the slice will change linearly with its distance from the starting point z. The thickness of the slice will be determined by the time step multiply the current cluster velocity. For each time step, we count all the coalescences happen in the slice and track the corresponding changes of the cluster concentrations. To count the collisions (which is just the coalescence number) happen in each time step, a model is proposed as
πΆπππππ πππ ππ’ππππ = πΆπππ π _π πππ‘πππ Γ π·ππππ’π πππ_πππ‘π Γ πΆππ’π π‘ππ_πππππππ‘πππ‘πππ_π Γ πΆππ’π π‘ππ_πππππππ‘πππ‘πππ_π Γ ππππ_π π‘ππ. (2.1) Cross section of collision is the sum of the cross sections of the two collision clusters (Fig.
2.1c). While diffusion rate comes from the mutual-diffusion theory, illustrating the diffusion rate of the carbon cluster in the atmosphere of the buffer gas, and is formatted as189, 190
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π·12= 0.001858π3/2βππ1+π2
1π2 ππ·
ππ122Ξ©π·, (2.2) where π1 is the mass of the buffer gas and π2 is the mass of the carbon cluster. π12 is the force constant for the mutual diffusion, and ππ· and Ξ©π· are regarded as constants. In our case, we have two kinds of the force constants where one is that for buffer gas-buffer gas (Fig. 2.1d) and the other for buffer gas-carbon cluster (Fig. 2.1e). Therefore, multiplying the collision cross section, diffusion rate and the time step, we get a volume that what a carbon cluster could sweep through during one time step. Subsequently multiplying the concentration of the two collided clusters, we could get the collision number. Collision model is the heart of this MC simulation.
Here using this model we actually take into account that influence of the buffer, which will be shown in the following simulation result. After updating the cluster concentrations according to the collisions happen, the program goes one step further, update the temperature, pressure, clustersβ velocity, etc., and then do the collision again and again. Once the slice evolves to the end of the jet or meets the termination criteria, the program stops.
Figure 2.1 (a) Schematic diagram of the fullerene synthesis. (b) Schematic diagram of an arc-jet MC model to simulate the fullerene formation. (c) Model of collision cross-section and diffusion rate to calculate collision number in each time step. (d-e) Models of collision cross-section for buffer gas-buffer gas and buffer gas-cluster, respectively.
Another important assumption of this model is the formation or the annealing of fullerene could only happen at an appropriate temperature zone (called fullerene formation period) from 3000 K to 1200 K. While at other temperatures, only cluster coalescence could happen while for fullerene formation and annealing the temperature is either too low or too high. This is because the initial model does not consider the temperature effect of the reactions, and we need to make it reasonable for the formation of fullerenes.
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Figure 2.2 (a) Fullerene yield using helium and argon as buffer gas at different pressures. Reproduced from Ref. 191. Copyright@2017, Elsevier Ltd. (b) Simulation result using the arc-jet model with an initial jet velocity of 30 m/s. Arc-jet model simulation results using (c) argon and (d) helium as buffer gas at different pressures and with different initial jet velocities..
With this simple model, we are already able to simulate some major results of the experiments.
With different types and pressures of the buffer gas, we plot Fig. 2.2b which resembles the results from experiments of Lamb et al. very much (Fig. 2.2a)191. Using helium as buffer gas, with the increase of the pressure, the fullerene yield will first increase dramatically, and reach the highest very soon. Later, the yield of fullerene will gradually decrease with the further increase of the pressure from 100 torr. While for argon gas, it is totally different. With the increase of buffer gas pressure, the yield is reducing all along rapidly. This different influence of the buffer gas is actually well explained by our collision model. As in equation (2.2), diffusion rate is inversely proportional to the pressure and the square root of mass of the buffer gas. Taking argon as buffer gas, the large mass of argon will refer to a smaller diffusion rate than helium, and with the increase of pressure, the diffusion rate is even smaller, which will certainly lead to the reduction of the fullerene yield since when the diffusion rate is too low, the collision number is also decreased (equation 2.1). While for helium, as the mass is small, at very low pressure, the diffusion rate becomes too high, leading to a large quantity of collisions.
Therefore, before entering the fullerene formation period, the clusters are already too large to be annealed to fullerenes, resulting in the lower yield (Fig. 2.3a top). With the increase of pressure, the diffusion rate decreases and therefore the collision number is reduced rationally, resulting in the increase of the yield (Fig. 2.3a middle). When further increasing the pressure,
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the diffusion rate is too low. With insufficient collisions, the final yield is lowered (Fig. 2.3a bottom).
Our model of collision well explains the role of buffer gas to the synthesis of fullerene, and agrees perfectly with the experimental results. It also describes the influence of various experimental conditions such as pressure (already discussed), temperature, volume, initial cluster velocity, etc. For example, for using either argon (Fig. 2.2c) or helium (Fig. 2.2d) as buffer gas, we tested the influence of the initial cluster velocity to the yield of the final products.
For both buffer gas, with higher initial cluster velocity, we observed the highest fullerene yield comes with lower pressure.
Figure 2.3(a) Cluster distribution at the fullerene formation temperature (~ 3000 K) with low (top), medium (middle), and high (bottom) buffer gas pressure. (b) Comparison of the final cluster distribution using the arc-jet model (red) and the mass spectra in experiments (black). Reproduced from Ref. 192. Copyright@2002, by Marcel Dekker, Inc. (c) Schematic diagram showing the energy profile of a reaction regarding the energy barrier and the reaction free energy.
However, the model is over-simplified with too many assumptions. For example, the most important assumption, that all collisions result in coalescence, is obviously not reasonable.
Whether a reaction could happen is determined by a reaction probability that is derived from the reaction free energy and the free energy barrier as we calculated. For a reaction whose energy profile is shown like in Fig. 2.3c, the reaction probability would be proportional to πβπΈπππ, while the reverse reaction probability would be proportional to πβ(βπΊ+πΈπ)ππ . And in reality, except for the collisions, in the fullerene formation, there are plenty of other reactions like
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fragmentation of the clusters or the transitions between different types of clusters happening.
This is why when we examine the final cluster distribution of the model, it is still far from the real experimental mass spectra of the synthesis product (Fig. 2.3b).
Therefore, to enrich our fullerene formation model, and to obtain more thermodynamic and kinetic information during the fullerene growth, we subsequently conducted both high accurate DFT calculations and MD simulations.