Chapter 1. Background and Method
1.3 Current theories and simulations on the formation of nano-carbon materials
Fullerene formation mechanisms: bottom-up
In typical mass spectra after fullerene synthesis using arc discharge or laser ablation method, usually high peaks of C60 and C70 fullerenes, as well as many smaller peaks of even-member fullerenes, are easily seen (Fig. 1.17a). Their structural features soon got attention in the fullerene field. Scientists found that, very similar to C60’s icosahedron skeleton, many fullerene obeys one very important rule named isolated pentagon rule (IPR), i.e. “a structure in which a pentagon is completely surrounded by hexagons is stable”116, 117. For example, Fowler et al.
discovered that the most likely candidate for the structure of C78 is one with isolated pentagons and a closed electronic shell (Fig. 1.17b)117. Subsequent studies showed that with the increasing of the fused pentagon pairs, the energy of the fullerene isomer increases118 as shown in Fig.
1.17c. Therefore, when in fullerenes there are fused pentagon pairs, structural evolutions, such as Stone-Wales (SW) transformation (Fig. 1.17d), will take place to reduce the energy. Thus, through frequent structural transformations at high temperature, most carbon clusters evolve to those particular isomers with high stability like C60 and C70, resulting in the relative abundance of these special peaks.
IPR rule is one of the critical principles to understand the typical mass spectra of the fullerene growth result. But how could the initial carbon vapor (including mostly C1 and C2 radicals) evolve to become complex icosahedral C60 structure is still an unsolved question, though there are many hypothesis and models for fullerene formation mechanism. At the beginning of fullerene formation process, small carbon fragments like C1 and C2 quickly aggregate to form chain structures (for Cn, n < 10) and monocyclic ring structures (for Cn, 10 <= n < 20), which is confirmed by a lot of experimental and theoretical works (Fig. 1.18b-c)119, 120. Also, it is demonstrated from the mass spectra that the final products are fullerene. However, what are the intermediates in between these two species and what are the processes from carbon chains and rings to the final carbon cages are still controversial.
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Figure 1.17 (a) Mass spectra of fullerene ions. Reproduced from Ref. 121. Copyright@2007, IOP Publishing, LTD. (b) Five isomers of C78 with isolated pentagons with views from three orthogonal axes.
Reproduced from Ref. 117 with permission from The Royal Society of Chemistry. (c) Energy differences of the three isomers of C70 containing 0, 2, and 3 adjacent pentagon pairs. Reproduced from Ref. 118. Copyright@2015, Springer Nature. (d) SW transformation. Reproduced from Ref. 122. Copyright@1992, Published by Elsevier Ltd.
Figure 1.18 (a) Theories for fullerene formation. Reproduced from Ref. 123. Copyright@1999, Korea Society of Mechanical Engineers. Optimized structures of Cn clusters (n = 3 - 24) calculated by (b) MBH/DFTB2 and (c) d2k methods. Reproduced from Ref. 120. Copyright@2015, AIP Publishing.
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Figure 1.19 (a) Hypothesis of the structural evolution for “Pentagon Road” of fullerene formation.
Reproduced from Ref. 124. Copyright@1988, Springer Nature. (b) Number of dangling bonds for hexagonal sheet and pentagon road of fullerene formation with the increasing of carbon atoms.
Reproduced from Ref. 125. Copyright@1991, Materials Research Society. (c-e) Hypothesis of fragmentation mechanism by the loss of C2, C4, and C6, respectively. Reproduced from Ref. 126. Copyright@1992, American Chemical Society. (f) Schematic diagram showing the initial steps in the formation of fullerene through bicyclic rings, and the following n steps to the way of fullerene (g). (h) Schematic diagram showing how large carbon ring zips up to form C60 fullerene. (f-h) Reproduced from Ref. 127. Copyright@1994, American Chemical Society.
“Pentagon Road” is one of the theories proposed. It was believed that for carbon clusters that have more than 20-40 atoms but are too small to make a closed fullerene with isolated pentagons, open graphitic cups are with much lower energy than any other structures (Fig.
1.19b)124, 125, 128. Therefore, in this way, basket-like structures, as the intermediates, grow into cages by the addition of small carbon units like pentagons (Fig. 1.19a). But up to now, there is no experimental evidence for the existence of these curved graphitic structures. Heath et al.
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proposed another “Fullerene Road”, trying to explain the synthesis of C60 molecules from small carbon clusters. Just like the possible mechanisms for the fragmentation of spheroidal carbon shells (Fig. 1.19c-e), in a reverse manner, a small closed fullerene cage could grow larger by the insertion of small carbon clusters like C2, C4 and C6 without the cage opening126. But this theory does not tell where these closed fullerenes come from. While in “Ring Coalescence”
theory, it was proposed that the carbon cages nucleate via the coalescence of large carbon rings instead of the addition of small carbon fragments (C2 or C3)127. Once we have mono-rings collide and form poly-rings, within few critical steps, there grows several polygons at the cross center which can be viewed as the nucleation (Fig. 1.19f). With next few steps, fullerene cages could form through further weaving of the left rings (Fig. 1.19g), and finally these large ring system could zip up to form a perfect fullerene cage (Fig. 1.19h).
Fullerene formation mechanisms: top-down
Figure 1.20 (a) Five trajectories showing how large fullerene cages shrink into smaller ones with fewer defects or dangling carbon atoms. Reproduced from Ref. 129. Copyright@2005, American Chemical Society. (b) High-resolution TEM images of the shrinking process of a giant fullerene C1300. (c) The annihilation of 5|7 defects by C2 removal in large fullerene cages. (b-c) Reproduced from Ref. 130. Copyright@2007, American Physical Society. (d-e) Simulated carbon cluster abundance maps with high and low isomerization rate, respectively. Reproduced from Ref. 131. Copyright@1996, Published by Elsevier B.V.
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Except for these bottom-up theories, there is also a top-down model called “Shrinking hot giant model”. MD simulations of Irle et al. showed that at high temperature (2000K to 3000K) the formed large fullerene cages are always with defects and dangling carbon chains, which could be further eliminated through annealing (Fig. 1.20a)128, 129. Later in 2007, Huang et al. observed the evaporation of the giant fullerene by real time microscope (Fig. 1.20b-c), which confirmed the previous simulation result130. However, either in simulations or experiments, the conditions for the formation of these giant fullerenes are different from those during the fullerene synthesis in arc discharge or laser ablation method, which means we still can not illustrate clearly how fullerene cages form from small carbon clusters.
Challenges in exploring the fullerene formation mechanism
Most theories failed to explain the fullerene formation mechanism because it is actually a very complex kinetic process which is far from thermodynamic equilibrium. Those structures predicted by static calculations is stable at ground state but not at the real synthetic conditions with high temperature, high pressure, especially with large temperature and pressure gradients.
In the early time, there were a few works on the arc-jet fullerene formation using Monte Carlo (MC) model. Alexandrov et al. considered the formation of different types of clusters using a kinetic MC model, including reactions of coalescence between various cluster, and in a sense simulated the cluster distribution very similar to the experimental mass spectra (Fig. 1.20d-e)131. However, their model was with the assumption of constant volume and temperature which as mentioned does not correspond with the real conditions. One thing to note is that in the fullerene synthesis, there is another important factor, the buffer gas. According to many experimental data, the type and pressure of the buffer gas could greatly influence the final yield. While the role of buffer gas has not been properly explained.
Go through the above introduction, it could be easily seen that the society is still quite ignorant about the fullerene formation. We do not know what the real carbon intermediates are during synthesis, and how carbon fragments evolve from small clusters to perfect fullerene spheres.
Fullerene growth is a big picture with complex environmental conditions including dramatic temperature and pressure change, with difficult physics in the jet turbulence, with possibly thousands of collisions and unknown reactions happening, and also with complicate thermodynamics and kinetics. To study the fullerene growth process, there is still a long way.
Carbon nanotube growth modes
Growth mechanism of CNT has been studied for more than two decades using either theoretical and simulation tools. Here in this section it will be briefly introduced about the basic theories on CNT growth, including how CNT nucleates on the catalyst surface, how the catalyst-CNT
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interface affects the growth thermodynamics, how the feedstock decomposition, carbon diffusion and incorporation influence the growth kinetics, etc. As mentioned in the previous section, according to the strength of particle-substrate interaction, there are base growth where the catalyst particle anchors on the support surface, and the tip growth where the particle leaves the substrate and moves as the CNT grows longer (Fig. 1.21a)76. Depending on the states of the catalyst particle, the growth of CNT could also be categorized into vapor-liquid-solid (VLS) mode where the particle is regarded as liquid at the growth temperature, and vapor-solid-solid (VSS) mode where the particle maintains its crystal lattice. The physical states of the catalyst will greatly affect the decomposition and the diffusion of the carbon species, and therefore influence the growth of CNT (Fig. 1.21b)132. In addition, according to the correlation between the diameter of particle and tube, CNT could grow in tangential mode where the particle and tube are of similar diameter size, or in perpendicular mode where the tube diameter is smaller than that of the particle (Fig. 1.21c)133. Therefore, the growth of CNT is actually very complex, and slight change of the experimental conditions would significantly change the growth of CNT.
To optimize the CNT growth and obtain high-quality and high-purity products, careful exploration on the CNT growth mechanism is necessary and very crucial. To simplify, in this thesis, we will mainly focus on the CVD growth mechanism of SWCNT
Figure 1.21 Schematic diagram showing different CNT growth modes including base and tip growth (a), VLS and VSS growth (b), tangential and perpendicular growth (c). (a-c) Reproduced from Ref. 132. Copyright@2018, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (d) Snapshots during MD simulation showing the nucleation process of CNT growth. Reproduced from Ref. 134. Copyright@2006, IOP Publishing Ltd. (e) Snapshots of MD simulation showing CNT nucleation process on Fe catalyst at 1500 K. (f) Number of polygon rings formed during the MD simulation in (e). (g) Schematic diagram showing the CNT nucleation mechanism via the formation of a pentagon. (e-g) Reproduced from Ref.
135. Copyright@2010, American Chemical Society.
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Carbon nanotube nucleation
Nucleation is one of the most important process during the CNT growth, and a lot of researches have been done to study the CNT nucleation on the catalyst particle. In 2003, Shibuta et al.
simulated the nucleation of a CNT on the Ni108 catalyst particle at 2500 K using classical MD136. A cap structure was formed when the carbon atoms inside the particle and on the surface could connect and form a hexagonal network. Ding et al. further studied the importance of supersaturated carbon concentration of the particle134. They argued that it is necessary to reach a highly supersaturated state of the particle for the subsequent formation of a carbon island during which the carbon concentration would gradually decrease (Fig. 1.21d). Therefore, only one CNT could be nucleated. Page et al. discovered, through density functional tight binding (DFTB) based MD simulations, that the nucleation of the cap starts by the “ring coalescence”
process just like in the fullerene formation theory we mentioned before while the first polygon formed at the crossing point of the rings is always a pentagon which they named it as “pentagon first” pattern (Fig. 1.21e-g)135.
Figure 1.22 (a) MD simulation results of CNT growth with different carbon-metal bond strengths. (b) Schematic diagram showing how carbon-metal interaction influences the CNT growth. (a-b) Reproduced from Ref. 137. Copyright@2007, American Chemical Society. (c) Optimized structures of cap on Ni(111) surface. (d) Energies of pristine CNT caps and CNT caps on Ni surface with different chiralities. (c-d) Reproduced from Ref. 138. Copyright@2006, Elsevier B.V. (e) Interface energies between CNT and catalyst surface calculated with MD and DFT, and the corresponding fitting. (f) Atomistic structures of CNTs on the catalyst surfaces. (e-f) Reproduced from Ref. 139. Copyright@2014, Springer Nature. (g)
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Top view of the (0 0 12) surface of W6Co7 nanoparticle. (h) Side views of the interfaces between CNTs of different chiralities and the (0 0 12) surface of the W6Co7 nanoparticle. (g-h) Reproduced from Ref.
95. Copyright@2014, Springer Nature. (i) CNTs on different positions of WC(110) surfaces with square, diamond and rectangle symmetry. Reproduced from Ref. 96. Copyright@2017, Springer Nature.
It is known that the structure of a CNT cap uniquely determine the chirality of the CNT if further chirality change in the continuous growth is not considered140. As in each cap there is always six pentagons according to Euler’s rule which is half of that for fullerene as mentioned before, therefore, the arrangement of these six pentagons actually defines the chirality of the subsequent CNT. In 2014, Penev et al. did an extensive search of all the cap energies and found that the energy differences for various CNT caps are relatively small compared to the energy differences for various CNT-catalyst interfaces141. Though the intrinsic cap energies are very similar, during the CNT growth, particular cap could be favored by their binding to the solid catalyst surface since the ab initio calculations showed that caps that have a lattice match with the catalyst surface are more stable than those who do not (fig. 1.22c-d)138. It was realized then that the CNT/cap-catalyst interface is an important factor in CNT growth.
Through DFT calculations and MD simulations, Ding et al. found that a strong carbon-metal interaction could efficiently support the growth of CNTs, just like that for the commonly used Fe, Co and Ni catalysts which leads to a strong adhesion between the CNT end and the catalyst particle (Fig. 1.22a-b)137. If carbon-metal binding (CNT-catalyst binding) is too weak (such as that for Cu, Pd, and Au), the CNT open end tends to close, resulting in the termination of the CNT growth (Fig. 1.22a-b). Artyukhov et al. later showed that armchair and zigzag tubes contact better with the catalyst surface (Ni or Co) compared to other chiral ones, and the energy of armchair-catalyst interface is more favorable than that of zigzag-catalyst interface (Fig.
1.22e-f)139. As these two kinds of tubes are more stable on the catalyst surface, they have more chance to be nucleated. Similarly, Yang et al. observed the chirality selectivity up to 92% of (12,6) tube in the CNT CVD synthesis using a very special W6Co7 nanoparticle95. They believed that it is because the surface of this nanoparticle has a perfect lattice match with the (12,6) tube (Fig.
1.22g-h). Later Zhang et al. was also able to synthesize CNTs with 90% (12,6) tubes or with 80% (8,4) tubes because of the high symmetry match between the catalyst surface and the tube rim (Fig. 1.22i)96.
Carbon nanotube elongation
After the nucleation, because of the thermodynamic stability, the initial distribution of the CNT chiralities is determined. However, this is not the final distribution yet. After a long time continued growth of CNTs, only the ones that can grow to a certain length would be counted, while those who can’t grow or are able to grow only a very limited length will be considered
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“dead”. So it is a combination of both the growth thermodynamics and the kinetics for predicting the CNT yield and the chirality distribution.
During the elongation of the CNTs, there are usually three steps to be considered: i) the decomposition of the carbon sources on the catalyst surface; ii) the diffusion of the carbon species to the CNT-catalyst interface, which is just the growth front; iii) the incorporation of the carbon atoms to the edge of the CNT wall. Mora et al. found that at low temperature, using the endothermic decomposition of CH4 gas as carbon precursor, the CNT growth is decomposition limited142. Only at a temperature above the decomposition threshold, the growth of CNTs would happen (Fig. 1.23a). Therefore in this case, step i) is the rate limiting step.
While Hofmann et al., using plasma-enhanced CVD method to growth CNT with Ni, Co, and Fe catalysts, demonstrated that step ii) could also become the threshold step143. They found a very low activation energy of 0.4 eV for the carbon species to diffuse on the catalyst surface through DFT calculations, which is actually much lower than CH4 decomposition barrier at 0.9 eV and C2H2 decomposition barrier at 1.2 eV.
Figure 1.23 (a) Schematic diagram showing that the decomposition of the feedstock limits the CNT growth. Reproduced from Ref. 142. Copyright@2008, American Chemical Society. (b) Schematic diagram showing the process of CNT growth at the growth front. (c) Minimum energy path (MEP) of the second step for the carbon incorporation at the CNT-catalyst interface. (d) MEP of the first step for the carbon incorporation at the growth front. (e) Energy profile for incorporating two carbon atoms into
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the CNT wall at the interface. (b-e) Reproduced from Ref. 144. Copyright@2011, American Physical Society. (f) A zigzag tube (n,0) can be viewed as a perfect crystal. While shifting by one or two Burger vector (g) the tube becomes chiral (n,1) (h) and (n,2) (i), respectively. (j) Free energy of the CNT during the growth. (f-j) Reproduced from Ref. 145. Copyright@2009 by The National Academy of Sciences of the USA. (k) Free energy during CNT growth within the addition of one hexagon. (l) Atomistic structure of (6,6) and (9,0) tubes on catalyst surface. (m) Density of different site types on CNT edges vs chiral angle. (n) CNT growth rate as a function of the chiral angle. (o) Predicted CNT distribution for CNTs with diameter of 0.8 nm and 1.2 nm. (p) Full CNT chirality distribution. (k-p) Reproduced from Ref. 139. Copyright@2014, Springer Nature.
Later in 2011, Yuan et al. studied the long overlooked step iii), the insertion of carbon atoms into the edge of the CNT wall, using DFT calculations144. They proposed that step iii) is with a significantly higher barrier of ~ 2 eV, which means the incorporation of carbon is actually a real rate limiting step for most experiments. As shown in Fig. 1.23b-e, there are two steps for inserting a complete hexagon to the edge of CNT at the CNT-catalyst interface, where the first step is to inserting one carbon atoms on the armchair site with a 1.02 eV barrier (Fig. 1.23d) and the second step is by inserting a second carbon to form the hexagon with a 1.20 eV barrier (Fig. 1.23c). So considering the entire process including the three steps shown in Fig. 1.23b, the total free energy barrier is 2.27 eV (Fig. 1.23e).
As demonstrated above that the rate limiting step for CNT growth is the incorporation of carbon atoms into the CNT wall, the growth rate of CNT could then be estimated by this incorporating rate, which has been proposed in 2008 by Ding et al., named as screw dislocation theory145. In the report, the authors argued that CNT can be viewed as having a screw dislocation along the tube axis, and therefore its growth rate is proportional to the burgers vector of this dislocation, which is just the chiral angle of the tube (Fig. 1.23f-j)145. With higher density of kinks at the CNT edge, the growth rate of the tube wall increases. Therefore, zigzag and armchair tubes are with lowest growth rate, and the chiral tubes are with much higher growth rate as demonstrated by the later DFT calculations (Fig. 1.23k-n)139. Now considering both the initial chiral distribution from the thermodynamics of CNT-catalyst interface and the growth rate from the kinetics of the carbon incorporation into the tube wall, the final CNT chirality distribution could be estimated (Fig. 1.23o-p), which agrees quite well with most experimental results for the abundance of (6,5) and (9,8) tube at the diameter range of 0.8 nm and 1.2 nm139.
Above prediction of the CNT abundance depends on the maintaining of the CNT chirality during the growth. However, it is possible to change the chirality of the tubes during the continued growth, i.e. by the formation of defects. For example, the addition of a pentagon- heptagon (5|7) pair would result in the change of the (10,0) tube to (9,0) or (9,1) tube (Fig.