Chapter 3. Growth of Single-Walled Carbon Nanotube on Metal Catalyst Particles
3.4 Carbon chain formation at growth front during CNT synthesis
3.4.2 Transformation between carbon chains and regular edges
Though sp2 carbon is more thermodynamics stable than carbon chains, most likely the elongation of CNT is a kinetic procedure. Therefore, it is very necessary to check the transformation process from carbon chains to CNT tube wall, and estimate that at normal carbon feeding rate if the process is able to complete.
Firstly, using CINEB method, we calculated the MEP of transforming a C2 carbon chain into a hexagon at the tube edge (Fig. 3.12a). In the initial state, a C2 carbon was attached to a kink site of the zigzag edge of the tube. The chain was bended to approach the tube edge. As the carbon from the chain end bonded with the carbon at the zigzag edge, a hexagon formed and the kind site propagated one step forward. The transition state of the reaction was found as well as the barrier of the transformation, which from the plot showing the energy profile during the
94
transformation (Fig. 3.12b) is 0.92 eV. To estimate the reaction time, we also considered the free energy barrier. At 1000 K, the barrier becomes a little bit higher to 0.99 eV.
Figure 3.12 (a) Initial, transition state, and the final structures of the transformation from a C2 carbon chain to a hexagon at the kink site on Ni(111) surface. (b) MEP of the transition from a C2 carbon chain to a hexagon at the kink site on Ni(111). (c) Minimum free energy paths of the transition from a C2 carbon chain to a hexagon at the kink site on Ni(111) at 0 K, 1000 K, and 1300 K, respectively.
Figure 3.13 Initial, transition state, and the final structures of the transformation from a (a) C3 or (c) C4
carbon chain to a hexagon at the kink site on Ni(111) surface. Minimum free energy paths of the transition
95
from a (b) C3 or (d) C4 carbon chain to a hexagon at the kink site on Ni(111) at 0 K, 1000 K, and 1300 K, respectively.
According to the transition state theory, the reaction time could be estimated by 𝜏~1
𝑘~ ℎ
𝑘𝐵𝑇exp (∆𝐺
𝑘𝐵𝑇), (3.1) where ℎ and 𝑘𝐵 are Plank constant and Boltzmann constant, ∆𝐺 and 𝑇 are the free energy and temperature of the reaction. With ∆𝐺~1 eV, taking 𝑇 as 1000 K, we roughly have the reaction time to be around 10 ns for this reaction.
Using the same method, we also checked the free energy barriers for the transformation reactions of longer carbon chains. Fig. 3.13a-b and c-d show the MEPs for C3 and C4 carbon chains at the kink site of the zigzag edge to transform into a hexagon with C1 and C2 carbon left at the kink site. The barriers are still around 1 eV and not increased much even at 100 K. For the transition of C4 chain, the barrier even decreases a little.
Figure 3.14 (a) Initial, the first transition state, intermediate state, the second transition state and the final structures of the transformation from a C3 carbon chain to a hexagon at the nucleation site on Ni(111) surface. (b) Minimum free energy paths of the transition from a C3 carbon chain to a hexagon at the nucleation site on Ni(111) at 0 K, 1000 K, and 1300 K, respectively.
Except for the barriers of the transition from chain to tube wall at the kink site, we also calculated the free energy barrier for a chain to transform into a new hexagon at the zigzag edge.
This is actually the nucleation of a new hexagon during the growth. Because we always have kink on the chiral tubes according to the dislocation theory, so it is not that important. But the
96
results show that the transition barriers for C3 and C4 chains at nucleation site on zigzag edge of the tube are 1.08 and 0.77 eV, still not very high (Fig. 3.14 and Fig. 3.15). Considering the temperature, the free energy barriers are 1.06 and 1.28 eV, while the barrier for long chains to transform is even lower (Fig. 3.14 and Fig. 3.15).
Figure 3.15 (a) Initial, the first transition state, intermediate state, the second transition state and the final structures of the transformation from a C4 carbon chain to a hexagon at the nucleation site on Ni(111) surface. (b) Minimum free energy paths of the transition from a C4 carbon chain to a hexagon at the nucleation site on Ni(111) at 0 K, 1000 K, and 1300 K, respectively.
So above calculations show that the reaction barrier of transforming a chain into a hexagon at the tube wall is around 1 eV, and therefore the reaction time round 10 ns. In normal MD simulations, because of the bond formation and breaking, a small time step, 0.5-1 fs, is usually used. This result in the small time scale of our MD simulations. For example, in classical MD, the longest time period we could do is around 100 ns; for DFTB method, it is around 500 ps;
for our DFT-MD simulations, it is even shorter, around 100 ps. But a transition reaction as we estimated would take 10 ns to happen at least. It means there is not enough time in the MD simulation for the chains to transform into hexagons. While in real experiments, the time scale is at least more micro second. It is enough for the chains to anneal into the tube wall.
Thus we conclude that the carbon chains in MD simulations is a consequence of the very short time scale, and the CNT-catalyst interface should be clean in real experimental condition as the
97
chains are able to be converted into hexagons in the CNT wall because of the macro time scale of annealing.
3.4.3 Evolution of carbon chains at growth front with different carbon feeding