Chapter 3. Growth of Single-Walled Carbon Nanotube on Metal Catalyst Particles
3.7 Chirality selectivity of SWCNT during nucleation
In Section 3.2, we studied the CNT nucleation on a metal catalyst particle using DFT based MD simulation. The method is extremely expensive and we cannot afford to use it for some critical issues requiring statistical analysis. For example, to understand the chirality control during CNT growth, it would need at least dozens of simulations to get a chirality distribution of the grown CNTs. Then further static DFT calculations stand out to play an important role.
3.7.1 Chirality assignment of SWCNT by the addition of the 6th pentagon into a graphitic cap with five pentagons
One of the top questions in SWCNT growth on metal catalyst particles is whether the chirality of the grown SWCNT is determined during the nucleation. It has been argued that a specific type of SWCNT with more stable cap-catalyst interface would be nucleated138, 221. Using the same catalystβa Ni55 liquid particle, we calculated with DFT the interface formation energies of SWCNTs with similar diameter but different chiralities (Fig. 3.39a). It was found that with the change of the chiral angle of the SWCNTs, the interface formation energy changes:
SWCNTs with low chiral angle are more stable on the catalyst than those with higher chiral angle. If the nucleation process is in thermal equilibrium or at least near the thermal equilibrium, the number of the SWCNTs with a specific type can be estimated by
124
π(π)~exp (βπΈπ(π)
πππ ), (3.5) where π is the chiral angle of the SWCNT, πΈπ(π) is the interface formation energy of the SWCNT with a chiral angle of π, ππ is the Boltzmann constant, and π is the temperature during growth. Thus, according to the calculation result in which (10,0) tube has the highest stability, (10,0) tubes should be with a great abundance in the grown SWCNTs, while in most experiments, the grown samples usually contain SWCNTs with different chiralities and each with similar abundances222, 223. This contradiction indicates that equation (3.5) is not able to describe the nucleation of SWCNTs. In other words, SWCNT nucleation is not a process near thermal equilibrium but governed by the kinetic laws.
Figure 3.39 (a) Interface formation energies of SWCNTs on liquid Ni55 particle. (b) The nucleation of a SWCNT by the addition of the 6th pentagon to a graphitic cap with five pentagons. (c) Schematic diagram showing the free energy evolution of the nucleation and the subsequent elongation of a SWCNT.
Reproduced from Ref. 207. Copyright@2018, The Royal Society of Chemistry.
To understand this kinetic process, we recall the scenery in the cap nucleation and our MD simulation. The nucleation starts with the formation of the first pentagon, and soon around this pentagon, there forms a sp2 carbon network by incorporating more pentagons and hexagons. It is when the number of pentagons in the network reaches 6 (if no other polygons except pentagon and hexagon), the formation of the cap completes and the elongation of the SWCNT starts.
125
Once the cap formation finishes, the chirality of the SWCNT is uniquely determined according to the positions of the 6 pentagons in the cap. In the center of Fig. 3.39b, we plotted the configuration of a SWCNT cap with 5 pentagons already arranged into the cap. One more pentagon to go to finish the formation of the entire cap. Depending on the site we added the 6th pentagon, we have then totally 11 different types of cap around the central configuration (Fig.
3.39b-b1 to b11), according to all the possible caps of the n+m=12 family SWCNT. This indicates that the addition of the 6th pentagon fully controls the chirality of the SWCNT despite the arrangement of the previous 5 pentagons, and the SWCNTs with different chiralities actually have the same probabilities to be nucleated if the incorporation of the 6th pentagon into the cap is random (Fig. 3.39b)207. Since a liquid catalyst has an isotropic surface, the probability of incorporating a pentagon into a cap should be equal.
According to crystal growth theory, the formation of a particle with its size smaller than the nucleus is a process near thermal equilibrium where the reactions are all reversible, and the formation of a particle with its size larger than the nucleus is a kinetic process where the reactions are all irreversible. Therefore, whether the size of a complete cap with 6 pentagons is larger than the size of a nucleus is the key to confirm the above discussion. If the above discussion is correct that the addition of the 6th pentagon is a kinetic process, we could imagine a picture like Fig. 3.39c. At the beginning of the nucleation, with the addition of pentagons into the cap, the cap formation energy increases very fast. When there are more pentagons in the cap, the energy increase becomes slower. At a critical size N*, the free energy reaches the maximum. This size N* should be smaller than the cap with 6 pentagons, since the formation of the 6th pentagon is, as expected, a kinetic process, and the free energy curve from the addition of the 6th pentagon to the elongation process should be with a negative slope.
3.7.2 Free energy during the nucleation of SWCNT
The kinetic incorporation of the 6th pentagon was demonstrated by our further DFT calculation of the entire nucleation process of a (11,1) SWCNT on a Ni55 catalyst particle. We gradually added C onto the catalyst surface and calculated the interface formation energies of the corresponding caps. The formation energies of the SWCNT caps are calculated by the following formula
Ef = ECNT@Ni55 β ENi55 β N * Ξ΅C, (3.6)
126
where ECNT@Ni55 is the energy of the whole system with SWCNT on the catalyst particle, ENi55
is the energy for only the Ni55 particle, and Ξ΅C is the energy of a C in SWCNT.
Figure 3.40 The formation energies of different size graphitic caps and different length (n,m) SWCNTs.
The images of the atomic structures of caps (I to VI, according to the addition of one to six pentagons in the cap) and short SWCNTs (VII and VIII) on metal catalyst are labeled with the number according to their formation energies in the free energy figure. Reproduced from Ref. 207. Copyright@2018, The Royal Society of Chemistry.
The data points of the formation energies are shown in Fig. 3.40 with blank circles and red stars.
The red stars with numbers labeled are the critical configurations with 1 to 6 pentagons incorporated, whose optimized atomic structures are presented below the formation energy plot with the corresponding number labeled. It can be seen clearly that the formation energy evolution of the cap can be fitted with a smooth curve and a horizontal line. And the linear part and the curve are well connected at the location of the 6th pentagon with their first derivatives exactly the same. We fitted the data points before the 6th pentagon with a curve as below
πΊ = 1
0.01558 +0.30038 Γ πβ
1 2
β 0.07689 Γ π, (3.7) with standard errors lower than 0.05 for all the parameters. And the data after the 6th pentagon was with simply linear fitting.
In experimental conditions of SWCNT growth, the Gibbs free energy for the cap formation is calculated as
127
G= Ef βN * ΞΌ, (3.8) where ΞΌ is the chemical potential difference between a C in feedstock and in SWCNT. As the chemical potential of a C in feedstock must be higher than that in SWCNT, we plotted the Gibbs free energies of the SWCNT nucleation with ΞΌ = 0.05, 0.1, 0.15, and 0.20 eV in Fig. 3.6, respectively. Fig. 3.40 shows that the maximum of the Gibbs free energy or say the nucleation point of SWCNT growth is shifted to N* = 52, 34, 23, and 18, whose sizes are all smaller than the size of the cap with 6 pentagons N* = 78. This calculation demonstrates the analysis shown in Fig. 3.39c that the incorporation of the 6th pentagon to the edge of the graphitic cap is beyond the nucleation size, and is a kinetic process with equal probabilities to grow into a SWCNT with a specific chiral angle.
3.7.3 Route towards selective growth of SWCNT
The above analysis and calculation verify the SWCNT nucleation is a kinetic process and on a liquid catalyst particle it is reasonable to grow SWCNTs with even chirality distribution.
However, on a solid catalyst, when the addition of the 6th pentagon at different position is not with the same probability as the local environments of the anisotropic catalyst surface are different, the formation of the 6th pentagon could be site-selective. Fig. 3.41a-g shows that on solid Ni55 surface, the addition of the 6th pentagon results in the different system energies.
Therefore, the formation of (10,1) and (6,5) tubes are with higher probability than the nearby zigzag and armchair SWCNTs as they lead to the local minimum in the system energy. In experiments, (6,5) tubes have been widely synthesized with great abundance in low temperature where the catalysts could still remain solid86. It agrees with our analysis perfectly. So to maintain a solid state of the catalyst particles could help to realize the selective growth of SWCNTs (e.g. W6Co7, WC, and MoC2)95, 96.
Another possible route to grow SWCNT with narrow chirality distribution is to change its chirality during the elongation process. It was argued that changing the growth temperature would change the chirality of a SWCNT224. In this process, the interface formation energy must play a very important role, and therefore, by frequently changing the growth temperature, the SWCNTs grown could be driven to the types with the most stable interfaces to the catalyst (Fig.
3.41h-n)225.
128
Figure 3.41 (g) The relative energies of (a-f) graphitic caps where the only difference of the caps is the location of the 6th pentagon. (h-j) Slight change of the temperature leads to the change of CNT chirality.
(k) Repeated change of the temperature can drive the (l) evenly distributed SWCNTs to (m) narrower distributed SWCNTs and finally to (n) the chirality control of the grown SWCNTs. Reproduced from Ref. 207. Copyright@2018, The Royal Society of Chemistry.