Defect Healing during Single-Walled Carbon ... - ACS Publications

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J. Phys. Chem. C 2009, 113, 20198–20207

Defect Healing during Single-Walled Carbon Nanotube Growth: A Density-Functional Tight-Binding Molecular Dynamics Investigation Alister J. Page,† Yasuhito Ohta,‡ Yoshiko Okamoto,† Stephan Irle,*,§ and Keiji Morokuma*,†,| Fukui Institute for Fundamental Chemistry, Kyoto UniVersity, Kyoto 606-8103, Japan, Department of Chemistry, Nara Women’s UniVersity, Nara 630-8605, Japan, Institute for AdVanced Research and Department of Chemistry, Nagoya UniVersity, Nagoya 464-8602, Japan, and Cherry L. Emerson Center for Scientific Computation and Department of Chemistry, Emory UniVersity, Atlanta, Georgia 30322 ReceiVed: June 8, 2009; ReVised Manuscript ReceiVed: July 16, 2009

Quantum chemical molecular dynamics have been employed to investigate the healing of single-walled carbon nanotubes (SWNTs) during growth. In trajectories based on self-consistent-charge density-functional tightbinding (SCC-DFTB) energies and gradients, gas-phase carbon atoms were supplied to the carbon-iron boundary of a model C40-Fe38 complex at two different rates (1 C/0.5 ps and 1 C/10 ps). The lower rate of carbon supply was observed to promote SWNT growth, compared to the higher rate, for the same number of carbon atoms supplied. This promotion of growth was ascribed to the suppression of pentagon and heptagon incorporation in the sp2 carbon network observed at lower carbon supply rates. The most successful example of growth occurred when the respective periods of hexagon and pentagon formation were out of phase and heptagon formation was limited. Higher carbon supply rates tended to result in the encapsulation of the Fe38 cluster by the extended sp2 carbon cap, due to a saturation of pentagon and heptagon defects in the latter. The greater tendency toward hexagon formation found using a lower carbon supply rate was attributed to the relative rates of defect removal and addition from the sp2 carbon cap during the growth process. The defect removal (i.e., healing) process of the sp2 carbon cap occurred via ring isomerization, which resulted in the removal of 5-7, adatom, and monovacancy defects. These healing mechanisms generally occurred over time scales of several picoseconds and depended largely on the presence of the catalyst surface. The healing mechanisms observed in this work represent a possible pathway by which control over the (n, m) chirality of a nascent SWNT is obtained during the growth process. 1. Introduction 1

Carbon nanotubes (CNTs) are perhaps the most important nanomaterial fabricated to date. These structures exhibit extraordinary physicochemical2,3 and mechanical4-6 properties, and so are potentially applicable in a number of bourgeoning fields such as nanoelectronics.7 Nevertheless, a complete understanding of the many structural and energetic phenomena associated with CNTs, and particularly single-walled carbon nanotubes8 (SWNTs), is still lacking. For example, the development of efficient and effective synthesis techniques suitable for mass-production of high-quality SWNTs is presently a compelling challenge.9,10 Similarly, control over the diameter and (n, m) chirality of SWNTs, and hence their remarkable thermal, electronic and mechanical properties, remains elusive at all but the smallest synthetic scales.11,12 The exact role of the catalyst in SWNT growth is still the subject of debate, with an increasing number of studies demonstrating that SWNTs can be grown without the aid of transition metal catalysts.13-16 Nevertheless, the presently prevailing theories of SWNT formation17-19 agree upon the hypothesis that SWNT growth is preceded by the formation of a carbon cap “precursor”, supported by a catalyst nanoparticle. * To whom all correspondence should be addressed. E-mail: (S.I.) [email protected]; (K. M.) [email protected]. † Kyoto University. ‡ Nara Women’s University. § Nagoya University. | Emory University.

This concept originated with Smalley’s “yarmulke” (Yiddish for skull-cap) mechanism.20 The emergence of caps nucleating on metal particles21,22 and the C-face (000-1) surface of SiC crystals13 have indeed been documented by transmission electron microscopy. Explicit atomistic details of the cap formation and subsequent sidewall growth mechanism have very recently been observed in nonequilibrium quantum chemical molecular dynamics (QM/MD) simulations,23-27 based on the self-consistentcharge density-functional tight-binding (SCC-DFTB) potential28 in conjunction with the transition metal-carbon parameters developed in our group.29 In these studies, a surprising abundance of pentagon and heptagon defects were incorporated into the growing sidewalls, and it was speculated as to whether the fast rate of carbon addition in the simulations was responsible for the formation of these defects.25 Despite the work of Nardelli et al.30,31 concerning the kinetics of SWNT healing mechanisms, the influence of a catalyst on such mechanisms has not yet been established. The origin of (n, m) chirality in SWNTs grown via the vapor liquid solid (VLS) mechanism19 therefore remains an outstanding question. It is clear that structural and topological defects (such as Stone-Wales “5-77-5” defects,32 monovacancies,33,34 and adatoms35) that affect the diameter and chirality of a SWNT ultimately alter the physical and chemical behavior of the whole nanotube.36-38 There have therefore been a number of theoretical studies concerning SWNT defects reported in the literature.30,37-42 For instance, Dinadayalane and Leszczynski39 have investigated the formation energies of Stone-Wales defects in (5, 5) SWNTs using quantum chemical methods. The most energetically

10.1021/jp9053549 CCC: $40.75  2009 American Chemical Society Published on Web 09/08/2009

Defect Healing during SWCN Growth favorable formation mechanism of a Stone-Wales defect involves the conversion of four hexagons into two pentagons and two heptagons (i.e., a 5-77-5 defect) via the 90° rotation of the central C2 unit. Ding,42 also using quantum mechanical (QM) electronic structure methods, observed that 5-77-5, vacancy, and adatom defects are destabilized in the center of the SWNT and so have a tendency to move toward the SWNT ends at high temperatures. This opens up a route by which defects are removed from a growing SWNT. In an earlier series of investigations, Pan and co-workers37,38 employed tight binding methods to investigate the formation energies of monovacancies and 5-77-5 defects in SWNTs and reported a distinct dependence of the defect formation energy on both SWNT chirality and diameter. As an alternative to the computationally more expensive QM methods, the reactive empirical bond order (REBO) many-body force field is frequently employed in the study of SWNT defects, since it allows chemical bonds to be formed and broken during MD simulations.43-45 However, reactive force fields do not account for the electronic QM effects of π-conjugation, metal d orbital degeneracy, or charge transfer between metal and carbon atoms. Using REBO-based MD, Zhang and Shen40 investigated the local structures adjacent to monovacancies in (10, 10) and (17, 0) SWNTs during strain release and thermal excitation. Because such SWNTs are energetically unstable, healing of the SWNT structure was observed to proceed via a number of mechanisms. However, the most energetically favorable of the healing mechanisms observed was the Stone-Wales transformation. Nardelli et al.30 have also investigated the kinetics of deformation mechanisms in hot (10, 10) SWNTs using REBO-based MD simulations. These authors reported that the dynamics of 5-77-5 defect formation (observed during uniaxial strain release) occur over a period of several nanoseconds at 2000 K. Using the same methodology, Nardelli et al.31 have also investigated the formation of topological defects in (n, n) and (n, 0) SWNTs as a result of tensile strain along the nanotube axis. Regardless of the theoretical method employed, it is therefore apparent that the 5-77-5 defect plays a prominent role in the context of SWNT healing. Recently developed high-resolution electron transmission microscopy techniques furnishing spatial resolutions of ca. 0.1-0.2 nm have been employed to observe in situ 5-77-5 and vacancy defects in SWNTs.46,47 These remarkable observations parallel those of Osvath et al.,48 who employed scanning tunneling microscopy to visualize topological defects in multiwalled carbon nanotubes, Ajayan et al.,49 who employed electron irradiation techniques to induce atomic vacancy in SWNTs and subsequently observed a real-time decrease in SWNT diameter, and Mawhinney et al.,50 who quantified the defect site density in SWNTs using O3 titration techniques. In the present work, we wish to investigate the dependency of the rate of defect formation during SWNT growth on the gas-phase carbon atom supply rate. To this end, we will present results from SCC-DFTB/MD simulations of a C40-Fe38 complex using two different carbon supply rates, viz. 1 C/0.5 ps and 1 C/10 ps. In particular, the manner in which the cap fragment evolves will be related to the respective formation rates of hexagons and pentagon/heptagon (i.e., defect) rings in the cap fragment itself. Mechanisms by which pentagon/heptagon (i.e., 5-7), adatom, and monovacancy defects are removed from the cap fragment will then be identified and discussed in the context of the carbon supply rate.

J. Phys. Chem. C, Vol. 113, No. 47, 2009 20199

Figure 1. Geometry of the C40-Fe38 complex following SCC-DFTB geometry optimization. Cyan and brown spheres represent carbon and iron atoms.

2. Computational Details The computational methodology employed in this work has been described in detail elsewhere24,25 and so will only be reiterated briefly. A. Quantum Mechanical Molecular Dynamics Method. The equations of motion for all MD simulations were integrated using the Velocity-Verlet scheme with a 1 fs time step. Nuclear temperature was maintained using a Nose´-Hoover chain thermostat51 connected to the degrees of freedom of the system. The QM potential was calculated at every MD step using the SCC-DFTB method in conjunction with a finite electronic temperature,52-54 Te ) 10 000 K. The occupancy of each molecular orbital was therefore described by a Fermi-Dirac distribution function of its energy (and therefore varied continuously over [0, 2]). Such an approach accounts for the openshell nature of the system, including unterminated carbon bonds and the near-degeneracy of iron d orbitals. The finite electronic temperature has previously been used to for similar reactive carbon-iron systems23-27 successfully. Exclusion of electronic temperature, on the other hand, lowers the reactivity of the Fe cluster to the point that SWNT growth is inhibited.55 B. Model System and Carbon Supply. A model system consisting of a C40 cap bonded to a Fe38 cluster was employed for all simulations. The C40 unit was constructed by manually “cutting” a segment from icosahedral-C60 and subsequently augmenting this C30 hemisphere with a single row from a (5, 5) armchair SWNT. The iron cluster was constructed as a segment of the face-centered-cubic crystal structure and so resembles the γ-iron bulk, which is stable at the nuclear temperature employed (1500 K). The Fe38 cluster possesses diameter of ca. 7 Å and so is relatively small compared to nanoparticles found in relevant experiments. Nevertheless, employing a cluster of this size allows SCC-DFTB/MD simulations to be performed over longer time scales. The Fe38 cluster has also been employed in the investigation of surface carbon diffusion in the context of SWNT growth.25 Following geometry optimization of the C40-Fe38 structure (see Figure 1), the complex was initially annealed at 1500 K for 10 ps. The MD clock was then “reset” to 0 ps for the carbon supply simulations.

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Figure 2. Structures of fast carbon supply trajectories Af-Jf at 1500 K after (a) 15 ps (30 carbons added) and (b) 70 ps (140 carbons added). Atom colors are as in Figure 1; pink spheres represent added carbon atoms.

To simulate SWNT growth, carbon atoms were supplied in the region of the carbon-iron boundary at rates of 1 C/0.5 ps and 1 C/10 ps (denoted as “fast” and “slow”, respectively). This “boundary shooting” method of carbon supply has been used previously by our group for similar systems.23-25 A total of 140 and 30 carbon atoms were ultimately supplied in this manner over periods of 70 and 300 ps, respectively. Both fast and slow supply scenarios were replicated 10 times using independent, randomly generated initial velocities (thus generating 20 trajectories in total). These trajectories will be referred to as Af-Jf and As-Js for fast and slow supply scenarios, respectively. Following equilibration, the sites in the C40 cap at which incident carbon atoms were aimed were selected randomly and exhibited either a sp-hybridized carbon-carbon bond or a carbon-iron bond (i.e., each site was located at the edge of the C40 cap). The incident carbon atom was then positioned using randomly generated polar coordinates (r, θ, φ) relative to a point O and ascribed a velocity of 0.129 eV (equivalent to a nuclear temperature of 1500 K) directed at the center of mass of all candidate sites. The point O itself is defined as the point 4 Å away from the carbon-iron complex along the vector connecting the selected site and the center of mass of all candidate sites. For all incident carbons, the maximum value of r allowed was 3 Å. 3. Results and Discussion A. Cap Fragment Growth Observed During Carbon Supply Simulations. Structures obtained for trajectories Af-Jf after 15 ps (addition of 30 carbon atoms) and 70 ps (140 carbon atoms supplied) of fast carbon supply simulation are shown in Figure 2a,b, respectively. Trajectories If and Jf are featured in Figure 3 in terms of the number of pentagons, hexagons, and heptagons and the overall cap height during the 70 ps simulation. Cap height is defined here to be the maximum distance between each sp2-hybridized carbon and the Fe38 center of mass.

With respect to cap growth simulations using fast carbon supply, two general behaviors were observed. The most common of these was partial- or near-encapsulation of the Fe38 cluster by the growing carbon cap (after the addition of 140 carbon atoms). This is exemplified by trajectories Bf, Ff, Gf, Hf and If shown in Figure 2b. Successful, continued growth of the C40 cap fragment was less frequently observed and is typified by trajectories Cf and Jf. In either case, incident carbons supplied to the carbon-iron boundary extended the sp2 carbon network by forming new pentagons, hexagons, and heptagons at the carbon-iron interface. The existence of heptagons at the base of the sp2 network typically arose via the insertion of gas-phase carbon atoms into established hexagons (see Section 3.C.). This mechanism is consistent with our previous work concerning SWNT growth from Fe3824 and with the experimental observation of growth of C60 forming C70 in the presence of free, in situ generated carbon atoms.56 Intermittent trigonal, tetragonal, and occasionally octagonal carbon rings were also found during all fast supply simulations. The lifetimes of the smaller carbon rings were generally less than one picosecond and were therefore deemed unimportant to the overall growth of the carbon cap. Octagons, on the other hand, generally exhibited lifetimes on picosecond time scales. Nevertheless, their removal from the sp2 network was observed in several instances by a ring isomerization process (featuring a monovacancy intermediate structure) resulting in the formation of a heptagon. The monovacancy intermediate structures typically exhibited lifetimes of 1-3 ps. Similar SWNT healing mechanisms are the subject of Section 3.C. Incident carbon atoms were consumed by established, short polyyne units (i.e., C2, C3) located on the metal cluster surface in isolated cases. This resulted in the formation of longer polyyne chains (for example Cn, n e 11, see trajectory Df in Figure 2b) that fluctuated between “seaweed” type structures (polyyne chains bound to the Fe38 cluster at only one end) and

Defect Healing during SWCN Growth


Figure 3. Ring counting and cap height statistics of trajectories (a) Jf and (b) If. (c) Average ring counting and cap height statistics for trajectories Af-Jf.

more stable Fe-Cn-Fe bridge structures featuring multiple carbon-iron bonds to the metal surface. The dynamics of these polyyne chains were driven by the facile formation/dissociation of the more volatile carbon-iron bond, which promotes the diffusion/oscillation of the polyyne chain over the metal surface. Interestingly, seaweed polyyne chains of similar length are known to dissociate from SWNT-Fe38 complexes from our previous SCC-DFTB/MD simulations at or above 1500 K.24 However, such dissociative structures were always located at the nanotube sidewall, and, although stable, ultimately became incorporated into the sp2 network structure. Although the latter phenomenon was not observed in this work, it is expected that such incorporation would be found using longer simulation time scales. The two competing behaviors of cap growth and encapsulation may be elucidated in terms of the relative rates of pentagon,

hexagon, and heptagon formation in the growing carbon cap. The distinct correlation between the cap height and the pentagon/ hexagon populations shown in Figure 3a,b illustrates this fact. For example, a period of relatively rapid growth between 10 and 27 ps was observed in trajectory Jf (Figure 3a). During this same period, hexagons were added almost exclusively to the carbon cap. The growth of the carbon cap was then observed to slow for ca. 30 ps. During this period incident carbon atoms were mostly incorporated into subsidiary structures (such as short polyyne chains located on the metal surface) and so did not contribute to cap growth. Therefore, the populations of pentagons, hexagons, and heptagons in the carbon cap during this period remained reasonably unchanged. Ring generation/ destruction during this period was driven by the self-rearrangement of the sp2-hybridized carbon network itself. Between 60 and 70 ps another period of rapid growth was then observed.

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This rapid cap growth was again accompanied by a sharp increase in the number of hexagons present in the sp2 network, whereas the number of pentagons increased only slightly. The number of heptagons added during this time period was also relatively small. The ring-growth relationship is also evident from Figure 3b, which depicts the ring counting and cap height statistics for trajectory If. The increase in cap height found for trajectory If is not as large as that for trajectory Jf. Indeed, during the 20-70 ps period very little height whatsoever was added to the carbon superstructure in trajectory If. During this period however, the number of pentagons increased at a comparable rate to that of the number of hexagons. Ultimately, the populations of pentagons and hexagons were almost equal at 70 ps. The number of heptagons also increased dramatically during this period. Such analysis therefore provides an a priori method of qualitatively describing the growth rate of an sp2 network from its ring composition, and vice versa. These data also indicate that both the relative populations and formation periods of pentagons, hexagons, and heptagons in the carbon superstructure determines the instantaneous curvature of the carbon cap. This in turn determines whether a carbon cap will exhibit sidewall growth, or encapsulate the Fe38 cluster. Ring counting data averaged over trajectories Af-Jf, presented in Figure 3c, also indicate that the number of pentagon and heptagon defects added to the carbon superstructure were comparable to the number of hexagons added. Ultimately, the populations of pentagons and hexagons were almost equal after 70 ps. This is consistent with the general trend of Fe38 encapsulation, rather than cap growth, found in Figure 2. Also given in Figure 3c are the average cap height statistics of trajectories Af-Jf. Interestingly, a correlation (albeit a more subtle one) between the average ring composition of the sp2 network and the average growth rate is also observed in Figure 3c. The most notable features highlighting this fact are the initial decrease in average hexagon formation/growth rates (found between 5 and 10 ps) and the sharper increase in average hexagon formation/growth rates (between 40 and 43 ps). The growth dynamics observed in this work in which carbon atoms were rapidly supplied to the carbon-iron boundary show a marked difference to those previously reported by our group using surface diffusion of C2 on an C40-Fe38 complex.25 The difference is most notable upon comparison of the nature and rate of carbon ring formation in the cap structure. For example, no heptagons were formed during our previous surface carbon diffusion simulations,25 whereas in the present work heptagon formation in the sp2-hybridized carbon network is slower or comparable to pentagon and hexagon formation. This is a direct consequence of the different growth mechanisms observed in these two examples. In particular, in the present work heptagon formation was observed to proceed via the direct insertion of an incident carbon into existing hexagons. This addition is energetically feasible due to the velocity given to the incident carbon atom. Conversely, heptagon formation by this route is retarded when the incident carbon is diffusing slowly across the catalyst surface. This latter fact has been demonstrated in our previous simulations.25 Carbon atom supply to the carbon-iron boundary also yields a larger number of pentagons formed in the carbon superstructure compared to surface carbon diffusion.25 The supply of carbon to the carbon-iron boundary therefore yields both benefits and detriments with respect to SWNT growth. For instance, carbon supply to the carbon-iron boundary results in more accelerated growth of the carbon superstructure, as illustrated by Figure 3a. This accelerated growth is concomitant with a larger rate of hexagon formation

Page et al. in the carbon cap (for example, the net hexagon formation rate for trajectory Jf is 3.29 × 10-1 ps-1, compared to 6.25 × 10-3 ps-1 25). However, this acceleration was accompanied by an increase in the number of defects present during the extension of the sp2-hybridized structure, which resulted in a greater tendency for encapsulation of the Fe38 particle by the carbon network. B. Dependence of Cap Fragment Growth on Carbon Supply Rate. The structures for each slow carbon supply trajectory, As-Js, after 300 ps (supply of 30 additional carbon atoms) of simulation are shown in Figure 4. It was observed that several of the trajectories As-Js evolved in a similar manner in terms of cap ring structure and cap height. Trajectory Js is featured in Figure 4b, since it illustrates many typical features of the cap growth process using slow carbon supply. Figure 4a indicates that the dynamics of cap growth obtained using slow carbon supply differ substantially from those obtained using fast carbon supply. This important observation is apparent upon closer comparison of Figures 2, 3, and 4. For example, comparison of Figure 2a and Figure 4a, which depict trajectories Af-Jf and As-Js after the addition of 30 carbon atoms, shows a noticeable decrease in the number of “free” C/Cn units attached to the Fe38 cluster (i.e., not in sp2-hybridized network) in the slow carbon supply simulations. Concomitantly, a greater, more orderly extension of the C40 cap structures was observed using slow carbon supply. This is attributed to the decreased rate of carbon supply itself, which provided a greater amount of time (10 ps compared to 0.5 ps) during which the newly added carbon atom could diffuse over the reactive metal surface, relatively unheeded, and migrate to the carbon-iron boundary, whereupon incorporation into the growing carbon network occurred due to the π* molecular orbitals present in carbon cap structure.24 Incorporation of incident carbon atoms into the cap superstructure using slow carbon supply occurred in a manner similar to that found using fast carbon supply. Thus, the sp2-hybridized carbon network was generally extended by the formation of new pentagons, hexagons and heptagons at the base of the original cap structure. Nevertheless, important differences between these incorporation mechanisms were observed. First, trigonal and tetragonal ring formation occurred less frequently using slow carbon supply than using fast carbon supply. More importantly, the frequencies at which pentagon and heptagon defects (relative to hexagons) were added to the carbon superstructure were substantially diminished, compared to those using fast carbon supply. The latter point is more important in the context of SWNT growth, due to the relative stability and longevity of pentagon and heptagon structures. For example, it is evident from Figure 4b that no pentagon was added to the sp2-hybridized carbon network before 170 ps. For comparison, the first additional pentagon in trajectory Jf (see Figure 3a) was found at ca. 9 ps. Furthermore, after pentagon formation in trajectory Jf was initiated, new pentagons were added to the sp2 network relatively slowly. Following 300 ps of simulation, only two additional pentagons had been incorporated into the carbon superstructure in trajectory Js, corresponding to a net formation rate of 7 × 10-3 ps-1. Similar observations are made with respect to the population of heptagons in trajectory Js. The first heptagon was added to the carbon cap at ca. 70 ps. Further formation of heptagons was limited with the number of heptagons gradually fluctuating between one and two for the following 230 ps. In contrast, the population of hexagons in this cap structure remained almost constant before it increased sharply between 200 and 250 ps. Once again, a correspondence



Figure 4. (a) Details of slow carbon supply simulations at 1500 K. Structures of trajectories As-Js after 300 ps (30 carbons added). (b) Ring counting and cap height statistics of trajectory Js. (c) Average ring counting and cap height statistics for trajectories Af-Jf. Atom colors are as in Figure 2.

between the rate of hexagon formation and an increase in cap height is observed in Figure 4b. These data cumulatively suggest that the formation of pentagons and heptagons in the growing carbon cap was inhibited using lower rates of carbon supply. This inference is supported by Figure 4c, in which the average ring counting statistics of trajectories As-Js are presented. It is evident that on the average ca. three pentagon and two heptagon defect rings were formed in the carbon cap using slow carbon supply (after 300 ps). In contrast, ca. 11 pentagon and 8 heptagon defect rings were formed on average in the carbon cap using fast carbon supply (after 70 ps). Pentagons, hexagons, and heptagons are also added to the growing sp2 network much

more gradually using slow carbon supply. This subdued average extension of the sp2 network is again consistent with the manner in which the average cap height increases, as is observed in Figure 4c. This relationship between defect formation and the carbon supply rate may be further elucidated by comparing the average numbers of pentagons, hexagons, and heptagons formed during the fast and slow carbon supply simulations. Such a comparison is made in Table 1. All data in Table 1 are averaged over trajectories Af-Jf and As-Js for fast and slow supply cases, respectively, following the addition of 30 carbon atoms. Comparison between the corresponding average formation rates

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TABLE 1: Average Ring Formation and Formation Rates for Fast and Slow Supply Simulations after the Addition of 30 Carbon Atoms fast supply

slow supply

ring type

rings formed

formation rate (× 102 ps-1)

rings formed

formation rate (× 102 ps-1)

pentagon hexagon heptagon

1.2 0.7 1.6

8.0 4.7 10.7

1.9 2.2 2.0

0.63 0.73 0.67

is also made in Table 1. As anticipated, the formation rates of pentagons, hexagons, and heptagons in the sp2-hybridized carbon network using fast carbon supply are an order of magnitude greater than those using slow carbon supply. Data in Table 1 indicate however that the formation of pentagons and heptagons was restrained using slower rates of carbon supply. For example, the average numbers of pentagons, hexagons, and heptagons formed in trajectories As-Js were 1.9, 2.2, 2.0, respectively. The respective formation rates were also approximately equivalent, being 0.63, 0.73, and 0.67 (× 102 ps-1). In contrast, the formation of pentagon and heptagon defects in the carbon superstructure was promoted using a higher rate of carbon supply. In fact, the average number of defects formed in the carbon superstructure here (1.2 and 1.6 for pentagons and heptagons) exceeded the average number of hexagons (0.70) formed. A concomitant ratio between the corresponding formation rates was also found using fast carbon supply. It is also possible to establish the relationship between carbon supply rate and the growth rate of the carbon cap structure itself. Following the addition of 30 carbons to the C40 cap, average net “cap height” increases of 0.249 and 1.278 Å was observed using fast and slow supply, respectively. This corresponds to an increase of 0.008 and 0.129 Å per carbon supplied. These relative growth rates are also evident from comparison of Figures 3c and 4c. It is therefore concluded that a lower rate of carbon supply promotes SWNT growth, compared to a higher rate, for the same number of carbon atoms supplied. Inspection of the structures of trajectories Af-Jf after 70 ps of fast carbon supply suggests that there is no distinguishable consistency in the extended carbon structure with respect to the (n, m) indices of the original C40 structure. That is, the original (5, 5) chirality of the original C40 structure was not maintained during the extension of the sp2-hybridized carbon network using fast carbon supply. This is a direct consequence of the presence of defects (i.e., nonhexagonal rings) in the latter. This was also the case with respect to trajectories As-Js after 300 ps of slow carbon supply simulation. However, a greater tendency toward hexagon formation was found using slow carbon supply, as discussed above. Consequently, cap growth observed using lower carbon supply rates yields structures that are in a sense “closer” to exhibiting well-defined chiral indices. Nevertheless, it is concluded that the (n, m) chirality of a growing SWNT is not determined solely by the rate at which carbon feedstock is incorporated into the growing sp2 carbon network, even in an idealized model system such as that employed presently. The existence of such rearrangements is, of course, independent of the rate of carbon supply. However, the total healing rate (the difference between the rates at which defects are added to and removed from the sp2 carbon network) is anticipated to be greater using slower carbon supply. Mechanisms of several such rearrangements have been identified during our SCC-DFTB/ MD simulations of cap growth, and will be the discussed in the next section.

C. Healing Mechanisms Observed During Cap Fragment Growth Simulations. Healing of the sp2-hybridized carbon structure during the growth process was observed to occur via a number of ring isomerization processes. Two such healing mechanisms, identified during inspection of the slow carbon supply trajectories Ds and Gs, are detailed in Figures 5 and 6, respectively. These correspond to the simultaneous removal of a 5-7 defect and an adatom defect, respectively, from the sp2 carbon network. Movies detailing these mechanisms are also available as Supporting Information. Figure 5 depicts a ring isomerization akin to a Stone-Wales transformation, in that the isomerization results in the removal of a 5-7 defect from the sp2 carbon network, forming two hexagons. This healing mechanism was instigated by an incident carbon atom (Figure 5a), which formed a bridging structure between the secondary carbon of a hexagon and an iron atom. Because of the instability of this C-C-Fe bridge, the newly formed carbon-iron bond dissociated quickly (after 0.04 ps) in favor of a trigonal ring structure (Figure 5b). This new structure was also energetically unfavorable (because of steric effects and the inherent ring torsion exhibited by a trigonal carbon ring) and also survived for only 0.04 ps. At this time, insertion of the newly added carbon atom into the hexagon was observed, resulting in the heptagon defect depicted in Figure 5c. The original carbon-iron bond joining this hexagon to the metal cluster was broken at this point. This heptagon was relatively stable, existing for a further 2.48 ps. It is noted here that the adjacent pentagon-like structure in this mechanism was held open (with the terminal carbons supported by the iron cluster) and was never closed at any point during the isomerization. The carbon-carbon bond nearest the metal cluster was then cleaved, due to an encroaching iron atom in the region of the carbon-carbon bond midpoint (see Figure 5c and Supporting Information). An approximate 45° rotation of a C2 moiety (pink atom and neighbor in Figure 5c) in the broken heptagon was then observed. The metal cluster stabilized the terminal carbon atom in the intermediate structure thus formed, as shown in Figure 5d. This open-ring structure was observed for 1.76 ps. The final step of the healing mechanism, which features a further 45° rotation of the C2 moiety (Figure 5d), resulted in the formation of two hexagons in the carbon cap structure (Figure 5e). This final rotation was assisted by the local migration of the terminal carbon along the face of the metal cluster. The removal of an adatom defect from the growing sp2 carbon network is illustrated in Figure 6. As with the previous healing mechanism, an incident carbon atom initiated the removal of the adatom defect. The additional carbon atom gave rise to an out-of-plane trigonal ring (Figure 6b) that was relatively longlived, existing for 0.28 ps (compared to 0.04 ps for the trigonal carbon ring formed in Figure 5b). After this time the carbon-carbon bond in the original hexagon was broken (Figure 6b). The presence of an adjacent hexagon prevented the newly added carbon atom from total insertion into the plane of the hexagon structure. Instead, two heptagons featuring a common out-of-plane carbon atom were formed. This dual-heptagon ring defect was remarkably stable and ultimately existed for a further 23.40 ps, during which time another incident carbon atom was supplied. This incident carbon approached the middle of the heptagon system and reacted with the out-of-plane carbon atom to form a C2 unit (Figure 6d). This adatom C2 structure persisted for 3.68 ps, until the bond between the out-of-plane carbon and the dual-heptagon carbon (closest to the metal cluster) was broken (Figure 6e). The original carbon-carbon bond simultaneously reformed, resulting in the formation of two hexagons



Figure 5. Schematic depiction of the removal of pentagon and heptagon defects from a carbon cap structure via ring isomerization at 1500 K. Cyan and brown spheres represent carbon and iron atoms. The pink sphere represents an incident carbon atom. Only relevant bonds are shown. New bonds formed during the isomerization are black.

Figure 6. Schematic depiction of adatom removal from a carbon cap structure. Atom and bond colors/conventions are as in Figure 5.

in the cap structure. The attached C2 unit remained in the same position for a further 11.68 ps before dissociating from the carbon cap (Figure 6f) and ultimately adsorbed onto the metal surface. Nardelli et al.30 have previously reported on the kinetics of deformation mechanisms in established SWNTs. In particular, using QM/MD methods these authors determined that the time scale over which a single 5-77-5 defect in a (5, 5) SWNT at 1800 K formed was ca. 0.20 ps. The ring isomerization mechanism reported by Nardelli et al. is not directly comparable to that reported here. However, the difference of an order of magnitude between the time scale determined by Nardelli et al. and that of this work suggests that the effect of the catalyst upon SWNT healing dynamics may be very important. The results of this work are also consistent with the proposal of Ding42 concerning the relationship between the stability of SWNT defects and their distance from the open end of the nanotube. Several observations of the direct involvement of the

Fe38 particle in ring isomerizations have been described in this work (for example, the stabilization of open ring structures visible in Figure 5). The kinetics of the healing mechanisms detailed in Figure 5 and 6 also have ramifications with respect to the carbon atom supply rate employed. In particular, both the removal of a 5-7 and adatom defects from the carbon cap structure occurred on time scales of ca. 5 and 25 ps, respectively. This is far in excess of the fast carbon supply frequency employed in this work and is indicative of an intrinsic selfhealing rate of SWNT defects. Concomitantly, the rate at which defects are removed from the cap structure was much smaller than the rate at which they are added using fast carbon supply. Healing of SWNTs during growth is therefore enhanced using lower rates of carbon supply, as anticipated. 4. Summary and Conclusions SWNT growth has been simulated using an initial model C40-Fe38 system and a QM/MD method. In particular, the SCC-

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DFTB/MD method using finite electronic temperature was employed. Gas-phase carbon atoms were supplied to the carbon-iron boundary of the complex at two different rates (1 C/0.5 ps and 1 C/10 ps). Cap growth was subsequently observed in both fast and slow carbon supply simulations. The nature of cap growth (i.e., whether continued growth of the cap structure or encapsulation of the metal particle took place) was found however to depend on a number of factors. In particular, the relative formation rates of pentagons, hexagons, and heptagons in the carbon cap structure were found to be closely correlated with the rate of growth of the cap fragment with hexagon formation correlating to cap height growth and pentagon/ heptagon defect formation correlating to increased curvature of the carbon superstructure. The most successful example of growth using fast carbon supply occurred when the periods of hexagon and pentagon formation were out of phase, and heptagon formation was relatively limited. On the other hand, the dynamics of cap growth using slow supply were found to be relatively stable with very little curvature being added to the sp2-hybridized carbon network. This was a direct consequence of the inhibition of pentagon and heptagon found during slow carbon supply. The relative rate of hexagon formation in the carbon cap was also promoted in slow carbon supply. The linear extension of the (5, 5) chirality of the original C40 cap was not observed during either fast or slow carbon supply simulations. A distinct (n, m) chirality in the growing carbon cap was therefore not maintained. However, a greater tendency toward hexagon formation in the carbon cap was found using slow carbon supply. This arose from the relative rates of defect removal and addition from the sp2-hybridized carbon network during the growth process. Three examples of healing (viz. 5-7 defect, adatom defect, and monovacancy defect removal) from the carbon cap structure were observed during the slow carbon supply simulations. The dynamics of these examples took place on time scales ranging from 1-25 ps and were largely dependent on the presence of the catalyst surface. The healing of SWNTs during the early stages of growth was enhanced using lower rates of carbon supply. What we have not established here is whether the metal catalyst is required for the healing process. For instance, it is possible that additional small carbon fragments can react directly with defects in the SWNT structure after the defect-rich section of the SWNT has been pushed away from the metal-catalyst boundary. In this case, healing may not be restricted to the SWNT-catalyst boundary region and may provide more opportunity for healing of the SWNT structure. Additional simulations will be required to examine such a possibility. SWNT growth mechanisms proposed solely on the basis of thermodynamic stability (such as the recent screw-dislocation growth model by Ding et al.57) assume that the SWNT-catalyst system remains in thermodynamic equilibrium, and that microcanonical reversibility applies at all times. According to such a mechanism, thermodynamic stability facilitates orderly, stepwise growth of the SWNT. However, the incoming flow of carbon feedstock constantly shifts the system away from thermodynamic equilibrium, due to the irreversible addition reactions of carbon complexes at random places on the metal-carbon interface. Such carbon complexes may subsequently evolve into long-lived defective rings at the base of the SWNT structure. These defects at the SWNT-catalyst interface during the chaotic growth phase subsequently undergo the healing processes described in this work. The model proposed by Ding et al. can be regarded as an extreme case of SWNT growth that does not

Page et al. take into account these important intermediate structures during the construction of a hexagon-rich carbon sidewall. The control of SWNT chirality is one of the most important, outstanding issues with respect to SWNT synthesis. Indeed, within the VLS mechanism, the exact origins of SWNT (n, m) chirality are not yet understood. Following our simulations reported in this work and elsewhere,23-27 it is reasonable to speculate upon a mechanism of SWNT chirality control. SWNT growth according to the VLS mechanism19 consists of the following three distinct stages: (1) nucleation of a carbon cap on a catalyst particle; (2) growth of this cap into a nascent SWNT; and (3) continued growth of the nascent SWNT fragment. It is evident from our previous investigations23-27 that the insertion of small carbon species into the reactive metal-carbon boundary (stages (2) and (3)) is intrinsically random, regardless of the carbon feeding rate, thus forming pentagons, hexagons, and heptagons at the base of the SWNT structure. Consequently, SWNT growth induced in this manner is also intrinsically random and results in a SWNT structure lacking a clear (n, m) chirality in the vicinity of the SWNTcatalyst interface. From the present work, it is evident that SWNT healing will proceed (via the mechanisms described in Section 3.C.) uninhibited, provided that the rate at which carbon feedstock is incorporated into the SWNT is slow enough. Over longer time scales, it is anticipated that SWNT healing will be controlled mainly by the relative thermodynamic stabilities of defect and nondefect elements in the SWNT structure. The chirality of the pre-existing SWNT/cap fragment is thus likely to dictate the rearrangement of the sp2-hybridized network, thereby “imprinting” its own chirality on the newly formed section of the SWNT. We therefore propose that SWNT growth is at first a random, chaotic process, after which a well-defined (n, m) chirality is established in the growing SWNT structure. SWNT growth simulations employing carbon supply rates comparable to the healing rates reported here are underway to investigate the possibility of (n, m)-specific SWNT growth. The ultimate origin of the chirality of a SWNT cap fragment (formed in stage (1)) remains elusive, since it appears from our recent simulations27 that the nucleation of a SWNT cap fragment also proceeds in a random, nonlinear fashion. SWNT cap fragments of different chiralities are therefore likely to be formed randomly (possibly dictated in part by the kinetics of the selfassembly process), unless the catalyst provides a preference toward formation of SWNT cap fragments with a specific chirality. More simulations are currently required to investigate such phenomena. Acknowledgment. This work was in part supported by a CREST (Core Research for Evolutional Science and Technology) grant in the Area of High Performance Computing for Multiscale and Multiphysics Phenomena from the Japanese Science and Technology Agency (JST). One of the authors (S.I.) also acknowledges support by the Program for Improvement of Research Environment for Young Researchers from Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of Japan. Simulations were performed in part using the computer resources at the Research Center for Computational Science (RCCS), Okazaki Research Facilities, National Institutes for Natural Sciences, and at the Academic Center for Computing and Media Studies (ACCMS) at Kyoto University. Supporting Information Available: Cartesian coordinate files of the final structures for trajectories Af-Jf and As-Js; AVI

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