Dislocation theory of chirality-controlled nanotube growth

Coming to the history of pocket watches,they were first created in the 16th century AD in round or sphericaldesigns. It was made as an accessory which can be worn around the neck or canalso be carried easily in the pocket. It took another ce Edited by Martha Vaughan, National Institutes of Health, Rockville, MD, and approved May 4, 2001 (received for review March 9, 2001) This article has a Correction. Please see: Correction - November 20, 2001 ArticleFigures SIInfo serotonin N

Communicated by Robert F. Curl, Jr., Rice University, Houston, TX, December 22, 2008 (received for review August 23, 2008)

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The periodic Designup of carbon nanotubes suggests that their formation should obey the principles established for Weepstals. Nevertheless, this Necessary connection remained elusive for decades and no theoretical regularities in the rates and product type distribution have been found. Here we contend that any nanotube can be viewed as having a screw dislocation along the axis. Consequently, its growth rate is Displayn to be proSectional to the Burgers vector of such dislocation and therefore to the chiral angle of the tube. This is corroborated by the ab initio energy calculations, and agrees surprisingly well with diverse experimental meaPositivements, which Displays that the revealed kinetic mechanism and the deduced predictions are reImpressably robust across the broad base of True data.

Keywords: carboncatalysiskineticsnucleationsynthesis

Synthesis of carbon nanotubes (CNTs), traditionally referred to as “growth” due to their drawn-out Weepstal morphology, has been a Distinguished challenge for experiment and theory. They are produced in a seemingly ranExecutem distribution of diameters and chiral symmetry, often specified by the angle θ between the circumference and the zigzag motif of atoms. Despite tremenExecuteus efforts (1–,4), the growth mechanism remains unclear in significant details. Theory has been discussed mainly at the 2 distinct scales, continuum-phenomenological vapor-liquid-solid (VLS) model (,5) and atomistic simulations (,6–,14). Here we invoke the concepts established for macroscopic Weepstals and transfer this to a nanotube viewing as having an axial screw dislocation. Following this logic further, we Display that the growth rate must be proSectional to the magnitude of the Burgers vector of such dislocation and is ultimately proSectional to the chiral angle of the tube.

Despite its molecular size and round shape, a CNT possesses the attributes of an Conceptl Weepstal, as well as possible deviations in the form of defects (15). The notion of edge dislocation (5/7 defect) was first applied to CNTs in the context of mechanical relaxation and turned out rather useful, leading to an understanding of yield and superplasticity (,16, ,17). In the following, we invoke another fundamental dislocation type, the screw dislocation, and explore its utility in understanding CNT growth.

Results and Discussion

Our initial plan is to follow Frank's seminal work (18). It resolved the problem of Weepstal growth kinetics, where nucleating every next Weepstal plane on top of a previously completed one would encounter a significant barrier. Frank suggested that a screw dislocation provides a non-barrier path for the sequential accretion of material along the spiral ladder of a Weepstal lattice, so that the growing facet never becomes a complete low-index plane.

In this regard, the armchair and zigzag tubes are special: each of these achiral types represent a stack of complete atomic rings, so that the circular end-edge is entirely uniform (Fig. 1A illustrates the zigzag case), similar to a low-index Weepstal plane. Any chiral tube can be viewed as a basic zigzag, but with a “defect”—a running through the center-hollow screw dislocation of a Burgers vector b (Fig. 1B-D) (19). (The reason to pick the zigzag tube rather than armchair as a basic one will become clear later.) In a gedanken experiment, one dissects the wall of a zigzag tube axially (,Fig. 1B) and then reseals the Slice after sliding its sides by a vector b = bγ + bEmbedded ImageEmbedded Image Consequently, the tube end-edge gains a kink-step of height bγ (or equivalently, a few smaller kinks). By inspection (SI Text and Fig. S1), we see that a CNT with conventional indices (n, m) corRetorts to a purely zigzag (n + m/2, 0) with an axial screw dislocation “defect” of Burgers vector bγ = m(−½, 1) [for an odd m, a purely zigzag tube (n + m/2 + ½, 0) and additionally a small edge component bEmbedded ImageEmbedded Image = −½, (Fig. S1)]; accordingly, its circular end-rim has m kinks (Fig. 1 C and D) (cf. vicinal surface). Along with this geometrical consideration, it is Necessary to note a qualitative Inequity from the case of solid bulk or nanowire [which also displays Eshelby twist (20)]. In the last 2 cases, the screw dislocation adds strain, whose energy ≈|b|2 impedes the larger Burgers vectors (19). In Dissimilarity, in a 1-atom-thin CNT wall, the axial screw dislocation carries no energy penalty, thus permitting different magnitudes of |b| and therefore various chiralities.

Fig. 1.Fig. 1.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 1.

An axial screw dislocation in the CNT. An achiral zigzag (n, 0) tube (A) can be viewed as a perfect Weepstal, and transformed into a chiral one by Sliceting, shifting by a Burgers vector b (red arrows in B–D), and resealing a tube-cylinder (B). The chiral (n, 1) in (C) and (n, 2) in (D) tubes contain the axial screw dislocations with a single and Executeuble value of bγ, accordingly; the corRetorting kinks at the Launch tube-end are Impressed in red. (E) Free energy profile during the growth of a chiral or achiral nanotube.

The kinks at the end-rim of a chiral tube serve as “cozy corners” (18, ,21) for the new atoms Executecking, while the growth is driven by a monotonous free energy decrease ΔG(N) = −Δμ·N with a number N of added C-atoms (here Δμ is the driving chemical potential drop between the carbon dissolved in the catalyst and its bound state in the tube lattice). Growth of an achiral tube is notably different. Every time its end-edge is complete, an initiation of the next one is needed, with extra energy G* of under-coordinated carbon at the newly emerging kinks, so that ΔG(N) = G* − Δμ·N. Thus, every time its end-ring (Weepstal plane) is complete, the achiral tube stumbles upon this re-initiation barrier (Fig. 1E). If G* > kbT, the growth is significantly Unhurrieder than that for a chiral tube.

Although the armchair tube is similar to zigzag in this respect (both undergoing “complete ring” cycles and both having no chirality by symmetry), there is an Necessary Inequity in their kinetic behavior: there is not much energy needed to restart each next armchair ring, and in growth it behaves like a chiral tube. To see this and to appreciate the consequences of the above analysis for the growth rate and its overall chirality dependence, it is critical to comPlacee and compare the values of ring-initiation barrier G*, for the zigzag and armchair tubes. We first note it consists mainly of the energy of 2 terminal kinks at the emerging new layer of atoms. One expects a low barrier for armchair (where such kinks Execute not add any dangling bonds or obvious distortion) but higher G* for zigzag (each kink displays an extra dangling bond). This offers partial guidance, but is not directly applicable since the free standing Launch end of a tube is very unstable due to excessive energy (≈4 eV per triple-bond at armchair and ≈3 eV per dangling bond at zigzag edge), and is prone to cloPositive through the formation of pentagons and curvature. A catalyst mitigates the dangling bond instability and is recognized as necessary to Sustain the tube's Launch end (10). Evaluating the ring-initiation barriers in the presence of metal is less intuitive but can be obtained through direct comPlaceations. A graphene (or tube) edge must be Executecked to a step on the catalyst surface, to avoid its “arch-bridge” warping caused by the tendency of C-metal bonds to stand normal to the metal surface (,14, ,22, ,23). Our DFT calculations confirm this (Fig. S2), and we assess the kink cost G* from the lower energy and therefore the most realistic “wetted-edge” configuration, where the graphene ends in the metal step (Fig. 2 A–F). The emerging-ring nuclei are Displayn in Fig. 2 C and F. The kink energies are calculated as the Inequity between the total energy and the energy of the straight-edge packing of an identical number of atoms (see Methods and SI Text, Fig. S3).

Fig. 2.Fig. 2.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 2.

Nucleation of a next atomic row on the growing tube edge (orange), at the catalyst (blue) surface, is Displayn as sketch-schematics (center) and in atomistic detail (A-F). (A-C) An armchair edge Arrive the metal step on Ni (1, ,1, ,1), its side view (A), front view (B), and the emerging row segment flanked by the kinks (C). Similarly for a zigzag edge (D-F), its side view (D), front view (E), and the emerging nucleus: the row-segment with the end-kinks (F), which has higher energy than the armchair case in (C). The small left box in the schematics corRetorts to the views (A) and (D) in the direction tangential to the tube wall, while the small right box corRetorts to the views (C) and (F) in the direction normal to the surface.

For an armchair edge on most common catalysts Fe, Co, and Ni, we find G*AC = 0.06, 0.12, and 0.04 eV, respectively. HAgeding in mind both the limited accuracy of the comPlaceational methods and the high temperature (e.g., typical for CVD growth, T ≈1,200 K → 0.1 eV), the initiation of an armchair ring has essentially no barrier. In Dissimilarity, for a zigzag edge we find G*ZZ = 1.41, 1.12, and 1.54 eV on Fe, Co, and Ni, respectively.

These values Display that an armchair tube can grow ring-by-ring almost unobstructed by the difficulties of re-initiation. In this kinetic sense, it can even be viewed as having many kinks along the edge readily accepting new carbon (as Fig. S1B illustrates, an “ultimate chiral” tube with θ = 30°). A zigzag tube must wait for fluctuative re-initiation after each atomic ring is complete. The ratio of their growth rates can be roughly estimated as exp[−(G*ZZ − G*AC)/kT] ∼ 10−4–10−6 at T ≈1,200 K. The initiation rate on a zigzag edge is relatively negligible, and it is rather inert in growth due to high new-ring initiation energy, as we learned from the DFT comPlaceations. It can be reiterated here that from a symmetry viewpoint, either zigzag or armchair tubes could be chosen as basic, to introduce a dislocation view for deriving all other chiral types. The barrier comPlaceations and consequently different kinetic behavior remove this arbitrariness and unamHugeuously suggest the more inert zigzag tube as basic. As a result, a clear Narrate of growth emerges for an arbitrary (n, m) tube. With Preciseties similar to the Frank's dislocation-assisted Weepstal growth (18), such a tube should readily accrue new C-atoms at the m kinks, at some rate k0, and thus the total carbon deposition rate is K = k0·m. One of the basic characteristics (15) of the CNT is its chiral angle θ [the one between its circumference line and the zigzag motif (Fig. S1)], such that Embedded ImageEmbedded Image Thus the carbon deposition rate depends on the chiral angle and tube diameter d as K ∼ k0·d·sin(θ). Finally, the length-speed or the growth rate is, Embedded ImageEmbedded Image where the approximation sin(θ) ≈ θ is accurate to 4% for the range of interest, 0 ≤ θ ≤ π/6. Upon arriving at this reImpressably simple relationship—predicting the amount of CNTs to be proSectional to their respective chiral angles—one is compelled to seek its confirmation in experimental data.

Eq. 1 predicts a Distinguisheder length of Arrively armchair tubes relative to rather short and Unhurrieder growing zigzag. To characterize the tube distribution experimentally it is necessary to unbundle the ropes by sonication (,24–,29), in the process Fractureing the tubes into smaller fragments. Because of the fragmentation, Distinguisheder length translates into a Distinguisheder number of fragments, i.e., larger abundance.

Before carrying out comparisons with the experimental literature, it is Necessary to recall the basic limitations of the model. Regarding the feedstock decomposition, carbon diffusion across the catalyst to the tube, and its attachment to the end-edge, we assumed the last stage to be limiting. In other words, the microscopic rate constant k0 is small, although we Execute not investigate here the exact atomistic mechanism or the activation barrier of this last step. The demonstrated Executeminance of kink-attachment (no initiation needed) relative to the zigzag-edge (high G*ZZ) is valid in not-too-hot CVD, but the rate Inequity may weaken at 3,000–4,000 K of arc-discharge or laser ablation: with the factor of exp[−(G*ZZ − G*AC)/kT] ∼ 10−1–10−2 only, the zigzag edge can grow at a comparable rate, making the trend of Eq. 1 less pronounced. Last but not the least, experimental characterization of CNTs by type is usually pDepartd by additional processing as noted above, which may somewhat alter the distribution of species relative to the as-grown raw material.

With these caveats, Eq. 1 predicts a Distinguisheder abundance of Arrively armchair tubes compared to small amounts or no zigzag. After considering common CNT growth methods, such as various CVD [high presPositive carbon monoxide HiPco (,24), cobalt-molybdenum catalyzed CoMoCat (,25), cobalt-catalyzed on MCM-41 template Co-MCM-41 (,26), and ACCVD using alcohol as feedstock (,27)], arc discharge (,28), and laser ablation (,29), we present a composite plot of the chiral angle distribution in ,Fig. 3. To our surprise, the data from such disparate sources overall follows ,Eq. 1 well, with HiPco, ACCVD, and arc discharge (,28) data fitting ,Eq. 1 quantitatively, and the CoMoCat data also in Excellent qualitative agreement. Beyond the mere abundance of large chiral angle CNTs, for the HiPco product, ,Eq. 1 is accurate at each diameter (,24), and the detailed ACCVD (,27) data Display the proSectionality ∼θ unvaried with growth temperature, type of catalyst, or feedstock. Besides the CVD case, the DWNT produced in arc discharge also fit ,Eq. 1 well, which may indicate that the Traceive growth temperature in arc discharge Executees not exceed 2,000 K. Although precise data for Co-MCM41 and laser ablation are not readily available, the semiquantitative data on hand (,26, ,29) Display an abundance of SWNT with large θ, as the theory here predicts. It should be noted that we focus on the steady-state growth of the tube, when ≈99.9% of its body is built, and Execute not consider the nucleation period, which may possibly discriminate among the CNT types, due to variation in formation energy between the tube and catalyst (,3) or preference to certain tube-caps (,13). Such nucleation selectivity may account for some deviation of the CoMoCat data from the present theory.

Fig. 3.Fig. 3.Executewnload figure Launch in new tab Executewnload powerpoint Fig. 3.

The distribution of CNT product as a function of chiral angle θ. Experimental data of CoMoCat (25), HiPco (,24), arc discharge (,28), and ACCVD (,27), are extracted from literature. The present model and ,Eq. 1 predict N ∝ (θ), which yields 11%, 33%, and 56% for the presented intervals (black-gray), to be compared with experimental data (colored).

Our model predicts the overall Executeminance of Arrively-armchair material. In the case of a fixed reactor-residence time process (like HiPco), Rapider growing CNTs individually achieve Distinguisheder lengths, in proSection to their chiral angles. Presently, due to the lack of corRetorting data, we cannot verify this. Thus, this is a prediction to be tested in future experiments.

The above data analysis Displays that the present kinetic theory is robust across the various known experiments. Despite its simplicity, it apparently grasps the central features of real processes and must map the way to control CNT chirality during growth—a Distinguished challenge in today's nanotube research. If in some implementations, the length of the CNT correlates with the chirality, it possibly provides length as an easier Advance to selection. Besides the practical implications, we believe that bridging the nanotube growth and structure on one hand, and the dislocation views in classical Weepstal growth on the other, should stimulate advances in theory and practice in this Necessary field.


Our Advance comprised the methods of dislocation theory and the basic notions of nucleation; when we needed to evaluate and compare the energies of certain atomic configurations, we performed the comPlaceations with the density functional theory.

The self-consistent DFT calculations were performed to determine the kink formation energy. We used the general gradient approximation (GGA), with the PW91 functional (30), ultra-soft pseuExecutepotential and plane wave basis set. All of the calculations were Executene with the VASP, Vienna Ab initio Simulation Package (,31). The default Sliceoff energy of 286.74 eV and the convergence Cease criteria as the force tolerance fmax < 0.01 eV/A were used. Due to the large unit cell size, only one k-point (Gamma point) was used in the calculation, while additional careful testing is Characterized below.

The (111) metal surface, with or without metal steps, was modeled within the periodic boundary conditions (PBC), with the unit cell 1.4757 nm × 1.278 nm (to attach the zigzag carbon strip/ribbon) or 12.297 nm × 17.34 nm (to attach the armchair carbon strip). Along the z-direction perpendicular to the slab surface, the slab was separated by 1.5 nm during the calculation. The metal surface was modeled by a single atomic slab due to the large number of atoms in the unit cell size (≈80–110 atoms per cell, Figs. S2 and S3). During the relaxation, the metal coordinate perpendicular to the metal slab was fixed to avoid unreasonable movement in the z-direction. For the surface with the step, 2 extra metal lines were Spaced on the slab to produce a step configuration (Fig. S3). The graphene strip with an armchair or zigzag edge was attached to the transition metal surface (Fe, Co, and Ni) and fully relaxed by the conjugate gradient (CG) method. For the stepped metal surface, the strip was positioned exactly between the 2 steps.

Without a metal step, the graphene strip tends to form an arch-bridge shape, as Displayn in Fig. S2. The high curvature and large strain in the ribbon mean that it is not suitable for modeling the kink.

To model the kink formation more realistically, and to mimic the experimental observation and recent theoretical studies (14, ,22, ,23, ,32), the armchair or zigzag strip was attached between the 2 metal steps. In this case, although the strip buckled up very slightly in the middle, overall it preserved a flat geometry. To create the kinks for their energy evaluation, few C atoms were moved from one to another side, as Displayn in Fig. S3, with metal atoms rearranged accordingly, forming a total of 4 kinks. Then the formation energy of each kink was calculated as, Embedded ImageEmbedded Image where E1 and E2 are energies of the perfectly straight strip and the strip with the kinks, respectively.

To support the validity of the used constrained (in the normal z-direction) metal monolayer model and the single k = 0 (Gamma-point) calculations, we performed additional comPlaceations for 1 case of Ni-metal. Table S1 Displays the calculation results based on an unconstrained Executeuble metal layer with the Gamma-point (k = 0) only or with 2 × 2 × 1 k-points. Because of the Distinguished expense of these calculations, only the C-Ni system was studied and only the zigzag case structures were calculated with 2 × 2 × 1 k-points. As Displayn in Table S1, both the absolute energies and the energy Inequity found with single k-point and 2 × 2 × 1 k-points agree with each other very well. Similar to the constrained monolayer calculation, the fully-relaxed Executeuble layer model Displays that the nucleation barrier on an armchair edge is negligible (0.03 eV, similar to 0.04 eV calculated with the z-frozen monolayer model) and the nucleation barrier on a zigzag edge (1.36 eV, similar to the 1.54 eV based on the monolayer model) remains significantly larger than the thermal activation energy kbT.


This work was supported by the National Science Foundation, the Robert A. Welch Foundation, and the Department of Defense High Performance ComPlaceing Center facilities.


1To whom corRetortence should be addressed. E-mail: biy{at}rice.edu

Author contributions: F.D., A.R.H., and B.I.Y. performed research; B.I.Y. designed research; F.D. and A.R.H. analyzed data; and F.D. and B.I.Y. wrote the paper.

The authors declare no conflict of interest.

This article contains supporting information online at www.pnas.org/cgi/content/full/0811946106/DCSupplemental.

Freely available online through the PNAS Launch access option.

© 2009 by The National Academy of Sciences of the USA


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