Why do nanotubes grow chiral?
WWhy Do Nanotubes Grow Chiral?
Vasilii I. Artyukhov, ∗ Evgeni S. Penev, ∗ and Boris I. Yakobson † Department of Materials Science and NanoEngineering, Rice University, Houston, TX 77005, USA (Dated: May 30, 2014)Carbon nanotubes (CNT) hold enormous technological promise. It can only be harnessed if one controls in apractical way the CNT chirality, the feature of the tubular carbon topology that governs all the CNT properties—electronic, optical, mechanical. Experiments in catalytic growth over the last decade have repeatedly revealed apuzzling strong preference towards minimally-chiral (near-armchair) CNT, challenging any existing hypothesesand turning chirality control ever more tantalizing, yet leaving its understanding elusive. Here we combine theCNT/catalyst interface thermodynamics with the kinetic growth theory to show that the unusual near-armchairpeaks emerge from the two antagonistic trends: energetic preference towards achiral CNT/catalyst interfaces vs.faster growth of chiral CNT. This narrow distribution is profoundly related with the peaked behavior of a simplefunction, x e − x . The broad interest in carbon nanotubes (CNT), unceasingsince their first clear observation [1], has been fueled by pos-sible technological applications derived from their unique fun-damental properties [2–4]. All of the latter are in turn deter-mined by the helical fashion of folding a tube, specified by thechiral angle χ between its circumference and the zigzag motifin the honeycomb lattice of atoms, with χ = ◦ and χ = ◦ for the achiral types, zigzag and armchair. Alternatively, a pairof chiral indices ( n , m ) is commonly used, the integer compo-nents of the circumference-vector [2]. In spite of such definingrole of chirality, most synthetic methods yield a broad distri-bution with mixed properties. To achieve control of the CNTtype remains a great challenge; what physical mechanisms de-termine the chirality distribution, and even why the nanotubesgrow chiral is still unsettled.Although the discovery paper by Iijima [1] has already sug-gested one key, connecting the tube ability to grow with thekinks at its end and the screw dislocation model, yet it tooknearly two decades till the first equation [5] related the speedof growth and chirality, R ∼ sin χ . It should further be use-ful to think, in hindsight, of the probable causes of such delay.Besides the difficulties of determining chirality in experiment,in theory it was ambiguous whether the chiral angle must bemeasured from the zigzag (as χ ) or perhaps from the armchair(as χ − ≡ ◦ − χ ) direction. While either choice appears validfrom pure symmetry standpoint, it changes the kinetic predic-tion to the opposite, and thus one stumbles upon an immediatecontradiction. Another diversion was due to simple thermody-namic argument that the lower energy of the tube edge, ratherthan its kinetic advantage of having kinks, must determine thedominant CNT type, pointing towards the armchair tubes, es-pecially ( , ) broadly discussed by Smalley et al. [6].This situation, together with recent advances in synthe-sis showing in several cases very narrow chiral distribu-tions [7–9], poses a compelling question of which factors—thermodynamic preference to lower energy, or kinetic prefer-ence of higher speed—play major role in defining the distribu-tion of CNT product. The true answer appears “both”, and theanalysis below shows how the subtle interplay of these phys-ical factors defines the more probable chirality choices. Inparticular, it explains why at lower temperature on solid cata- lyst particles the yield is peaked near armchair type ( n , n − ) ,never exactly armchair, although quite close.The evolution of chemical vapor deposition (CVD) tech-niques and chirality characterization methods [10] has led toimprovements in chiral selectivity [7–9, 11–16], eventuallyreaching >
50% fraction for a single CNT type and ∼ nucleation occurs, withprobability N n , m of certain chiral type. It is followed by thesteady carbon accretion by each tube, with its growth rateR n , m . After some time, the fraction of the tubes of chiral-ity ( n , m ) , i.e., their relative abundance A n , m in the accumu-lated material, is determined by the product of both thesefactors [32] as A n , m = N n , m · R n , m . Using instead of the chi-ral indexes the tube diameter d and chiral angle χ one has, A ( χ , d ) = N ( χ , d ) · R ( χ , d ) . Below we explore the physicalmechanisms defining the right-hand side, in particular the caseof solid catalyst with rigid shape, which yields sharp chiral an-gle selectivity in A ( χ , d ) , as empirical evidence suggests.During nucleation , as carbon atoms attach to a nascentCNT nucleus, adding new hexagonal and pentagonal ringsto it, the chirality of a CNT becomes permanently “lockedin” when the final 6 th pentagon is added to the hemisphericalcap. From this 6-pentagon nucleus a cylindrical CNT struc- a r X i v : . [ c ond - m a t . m t r l - s c i ] M a y ± ± chiral angle  d a b c def Fig. 1 . Continuum model of the CNT–catalyst system.
Schematicrepresentation of ( a ) achiral, ( b ) multiple-kink chiral, and ( c ) single-kink chiral CNT on a flat substrate. Unrolled CNT–substrate inter-faces for ( d ) two-kink and ( e ) single-kink nanotubes show the nan-otube tilt off the vertical, reducing the edge-substrate gap; the whitedot in ( c ) and ( e ) marks the contact point. The abundance distribu-tions A ( χ ) ∼ χ e − χ computed as the product of nucleation (dotted)and growth rate (dashed) terms are shown in ( f ) for near- Z (blue)and near- A (red) chiralities. The inset illustrates a nascent CNT ofdiameter d on a solid catalyst. ture can further grow by adding only hexagons, in a periodicfashion. The free energy of the critical nucleus contains twocontributions, G ∗ = G cap + Γ . The first one, G cap , originatesfrom the “elastic” energy of cap per se and does not dependon χ [33]. The second term Γ represents the contact inter-face between the sp -carbon lattice edge and the metal cat-alyst, and does contain chirality dependence since the edgeenergy γ ( χ ) varies with the crystallographic orientation and Γ ( χ , d ) ≡ π d γ ( χ ) . Whereas previous studies on edge energet-ics [34] assumed either vacuum or a liquid-like catalyst thatfully adapts to the edge shape, chiral-selective CVD growthis usually reported at comparatively low temperatures [8, 9]when the catalyst particle is solid [9]. Accordingly, the metalside of the interface is rather a rigid atomic plane, and thestructure of this interface affects both the energy of the nu-cleus and the subsequent insertion of new C-atoms duringgrowth.Before discussing the details of atomistic study, it is use-ful to explore the key ideas in terms of simpler continuummodel , which not only offers valuable insight but is even ableto make accurate overall predictions. In Fig. 1 inset, the CNTis in contact with the catalyst which is represented locally as acontinuous plane corresponding to an atomic terrace in a solidparticle. The CNT is also continuous, but the kinks aroundits edge are retained according to the tube chirality,
Fig. 1a–c . These kinks cause the gaps between the substrate and theCNT, shown in
Fig. 1b–c for ( n , ) and ( n , ) tubes, with anassociated energy penalty, relative to the tight contact in caseof achiral tube in Fig. 1a . For the hexagonal lattice of CNT the two fundamental achiral edges—armchair ( A ) and zigzag( Z )-form tight low-energy contacts. The interface energy forchiral tubes is higher, roughly in proportion with the numberof kinks, which raises linearly with χ for near- Z tubes, or with χ − for near- A tubes. In other words, γ ( x ) ≈ γ + γ (cid:48) · x , where x is the angular deviation from the achiral direction: near the Z -type x = χ , γ = γ Z ≡ γ ( ◦ ) , and γ (cid:48) = ∂ γ / ∂ χ | χ = ◦ , or nearthe A -type x = χ − , γ = γ A ≡ γ ( ◦ ) , and γ (cid:48) = − ∂ γ / ∂ χ | χ = ◦ .Since γ ( x ) is largest in the intermediate range of χ ≈ ◦ ,such tubes are unlikely to nucleate, and one should focus onjust the neighborhoods of Z and A chiralities. Then we write(omitting for brevity the k B T factor, wherever obvious): N ( χ , d ) ∝ e − G ∗ ∝ e − π d ( γ + γ (cid:48) · x ) . (1)The essential result here is that the nucleation probability fallsrapidly as e − β · x with chiral angle x and β = π d γ (cid:48) / k B T . A dis-tinction for single-kink cylinders, representing the nanotubes ( n , ) and ( n , n − ) , should be noted. Their symmetry allowsthem to tilt in the vertical plane, improving the interface con-tact. Fig. 1d–e illustrates it by “unrolling” the CNT–substrateinterface area for two-kink and single-kink tubes. In the lattercase, the effect of tilt leads to a reduction of the tube–substrateseparation, appearing as a sinusoid along the circumference asshown in
Fig. 1e , enabling a substantial closure of the gap be-tween substrate and tube edge, recovering up to as much as70% of the energy penalty according to our estimates.For the growth rate term R ( χ , d ) we augment the screwdislocation model [5] by including the kinks created by ther-mal fluctuations on A and Z edges [35], and accounting for theenergy penalty ∼ / d from the wall curvature. In the liquid-catalyst model, when the metal adapts to the CNT edge with aone-to-one termination, calculations suggest that the cost E A to create a pair of kinks on an A edge is zero, and consequently R ∝ χ [5]. However, on a solid surface, creating a pair of kinksdestroys the perfect contact between the CNT and substrate,costing energy. Therefore E A has a noticeable magnitude, andthe dependence becomes bimodal with minima at the A and Z ends of the chiral angle range, and a maximum at the magicangle of 19 . ◦ [35, 36]. The final expression, linearized nearthe A and Z bounds of chirality reads as follows, R ( χ , d ) ∝ π d e − C / d ( x + e − E ) , nearly as ∝ x , (2)where C = . · ˚A /atom is the bending rigidity ofgraphene [37]. The term x in parentheses corresponds tothe density of geometry-imposed kinks, proportional to thevicinal-edge angular deviation from the main achiral direc-tion, and the term e − E (typically small) represents the addi-tional fluctuational kinks. The free energy barriers for the ini-tiation of a new atomic row on A or Z edge are E = E Z nearthe Z -type where x = χ / √
3, or E = E A near the A -type where x = χ − .Multiplying together the nucleation and growth termspresents the key to understanding the observed selectivity fornear- A chiralities [9]. At a given diameter, A ( χ ) = N ( χ ) R ( χ ) ∼ χ e − χ , (3) ° ( e V / Å ) chiral angle  ( ± ) d ' d ' ab (9,0) (9,1) (6,5) (6,6) Fig. 2 . Chirality-dependent CNT–catalyst contact energies, gov-erning the nucleation. ( a ) Interface energies calculated with MD(circles and triangles) and fitted with analytical expression (solidline) for two CNT sets (inset; dashed arcs denote the range of diam-eter variation in each set). Static DFT calculations on Ni and Co arealso shown. Open and filled symbols denote regular (hexagonal) andKlein Z edge structures—see sample atomistic structures ( b ). Thedash-dotted line corresponds to liquid catalyst case. In ( b ) the blueand red atoms highlight the Z - and A -edges, respectively. a function with a sharp peak near zero (or near 30 ◦ ). This is theessential result of our continuum consideration . Fig. 1f illus-trates this peaked distribution character. The two distributionsfor near- Z (blue) and near- A (red) chiral angles are plotted as-suming equal interface energies and growth barriers: γ A = γ Z , E A = E Z . However, if either A or Z has a lower energy, theopposite peak distribution is additionally penalized by e − ∆Γ ,with ∆Γ = Γ Z − Γ A on the order of eV, and then one wouldexpect to observe only one side of the distribution. Thesecontinuum-model predictions turn out to be remarkably ro-bust. To see this, below we present the atomistic calculationsof the relevant quantities, and proceed to simulate exampleCNT type distributions. Atomistic computations were performed using a flatNi ( ) slab to represent the solid catalyst. We used aclassical force-field MD sampling complemented with staticDFT computations. MD calculations were performed us-ing the canonical ( NV T ) ensemble with the ReaxFF forcefield [38, 39] as implemented in the LAMMPS simulationpackage [40, 41]. DFT calculations were performed withthe local spin density approximation using the Q
UANTUM
ESPRESSO package [42].To investigate the chiral selectivity of nucleation , two setsof CNT are chosen, with d ≈ . . ( , ) and ( , ) CNT.
Fig. 2a shows
012 0 5 10 15 chiral (9,0) (9,9)(6,6) ¢ G ( e V ) N ZK A s i t e s / l a tt i c e pa r a m e t e r ± ± ± d =
10 Å8.5 Å7 Å r e l a t i v e g r o w t h s peed chiral angle  ab cd (6,6)(9,0) Fig. 3 . Chirality-dependent growth rate of CNT. ( a ) Free energyprofiles during the growth of a new ring of hexagons on (red, orange) A and (blue) Z edges as a function of number of added atoms N .The green line corresponds to barrierless chiral edge growth. ( b ) Theatomic configurations after first dimer addition, N =
2. ( c ) Lineardensity of different site types on CNT edges as a function of chiralangle. ( d ) The resulting CNT growth rate as a function of chiral anglefor several diameters (inset shows the effect of thermal kinks). the calculated CNT–substrate interface energies. The atom-istic structures for the ( , ) , ( , ) , and ( , ) CNT are shownin
Fig. 2b , where the tilting of the ( , ) and ( , ) CNT toreduce the interface energy is seen clearly. We found thatfor the smaller-diameter set, for near- Z edges a Klein struc-ture with dangling C atoms [43] is favored over the standardclosed-hexagons, whereas the larger-diameter set shows littlepreference either way. All data display the same qualitativebehavior conforming to the above discussion, and are gener-ally in good quantitative agreement. Both bounds of the chiralangle range (achiral CNT) are energy minima, and the energyis higher for Z than for A tubes. The curves show the fit of γ ( χ ) using the analytical expression from earlier work [34]for solid and liquid-like cases.Our computations pertaining to the growth kinetics aresummarized in Fig. 3 . We build upon our earlier approachfor graphene [35], adapting it for the case of CNT.
Fig. 3a shows the energy changes with the addition of a new rowof carbon atoms, dimer by dimer, for ( , ) , ( , ) and ( , ) CNT. All three curves depart from the “nucleation to kink-flow” scenario of graphene, the reason being the constantlychanging tilt angle of the CNT. However, both A curves (red,orange) show essentially the same height for the first dimeraddition and the same maximum height (closer to the end).The Z curve bears the same qualitative character, having aninitial and a final maximum. The maximum height of eachcurve determines the free energy barrier that needs to be over-come for successful addition of each new row of hexagons, ∆ G A ≈ . ∆ G Z > ( , ) CNT yield multiple intermediate structures with topo- ab – z i g z a g – – a r m c ha i r – (cid:72) n , n (cid:76) (cid:72) n , 0 (cid:76)(cid:72) n ,1 (cid:76)(cid:72) n , n (cid:45) (cid:76) Fig. 4 . Predicted CNT type distributions. ( a ) Distributions cal-culated directly based on MD computations for two CNT sets ( d ≈ . b )Full ( n , m ) distribution based on an analytical fit to MD interface en-ergies. For all plots the temperature was artificially set to about threetimes the typical experimental value to make the heights visible. logical defects of energies lower than perfect hexagonal struc-tures, an additional complication for Z -CNT growth). Theseare the terms that penalize the pure A and Z tubes, comparedto the chiral ones (green line in Fig. 3a ), by an additional fac-tor ∝ e − ∆ G and thus effectively remove them from the productdistribution, despite their favorable cap-nucleation energies.The atomic configurations for the first dimer addition to the ( , ) and ( , ) CNT are shown in
Fig. 3b . Fig. 3c shows theconcentrations of different site types— Z , A , and K (kink)—asa function of χ . Thin black lines show the intrinsic, topo-logically required values. Finally, Fig. 3d shows the CNTgrowth speed. Higher temperatures favor kink formation andpromote the growth of zigzag and armchair CNT. However,under realistic conditions, when k B T (cid:28) ∆ G , the growth rateof achiral CNT is negligible. Among different diameters d ,the curvature of the wall penalizes insertion of C atoms intosmall-diameter CNT, as in Eq. 2.We now have all the ingredients to calculate the relativeabundance of different CNT types. The distributions for thetwo CNT data sets in Fig. 4a both show a strong predomi- nance of ( n , n − ) near-armchair CNT. The selectivity of dis-tributions is actually so strong that one has to increase the tem-perature to T = T =
900 K, both show effectively single peaks for ( , ) or ( , ) CNT. Further, one can use either data set to computea general ( n , m ) distribution through interface energy fitting.An example is shown in Fig. 4b . The peak in diameter dis-tribution results from the competition between the interfaceenergy, favoring smaller d in Eq. 1, and a prefactor due to theconfigurational entropy of CNT caps [33, 44], favoring largerdiameters [2], which scales approximately as ∼ d [45]. In areal CVD experiment, there will be additional constraints on d , from the size of catalyst particles. Then the product dis-tribution will be a slice of Fig. 4b with prominent ( n , n − ) peaks, such as those shown in Fig. 4a .By similar logic, our theory suggests a possibility to highlyselectively achieve the near- Z ( n , ) CNT, if a catalyst favors Z interface over A . While ( n , n − ) are always semiconduct-ing, the ( n , ) series contains all three CNT families (metal-lic and two semiconducting). Then, a control of diameterwould allow a selective synthesis of CNT of either conduc-tivity type. Moreover, when speculating on a possibility ofcatalyst-template exactly matching a certain ( n , m ) tube, welearn here that this would more likely favor the one-index-off tubes ( n , m ± ) , to allow for rapid kinetics at the cost of some-what higher energy of the contact and nucleation.We can also compare the simulated distributions to a liquidcatalyst model (dashed-dotted line in Fig. 2a and the kink for-mation energy E A = Fig. 4a as hollow bars and display much greater presence of armchairCNT in the overall broader distribution. In reality, irregularand highly mobile structure of liquid catalyst may flatten theenergy landscape in
Fig. 2a and thus further broaden the dis-tribution. If E A >
0, the fastest-growing tubes have χ = . ◦ ( Fig. 3d ), which corresponds to ( m , m ) CNT. Finally, with χ -unbiased nucleation probability and E A →
0, one recoversthe proportionality result, A ∝ χ [5].In summary, the analysis above shows that the kinetic andthermodynamic aspects of CNT growth must be consideredconcurrently. The growth kinetics is aided by the kinks atthe tube edge and thus favors the chiral types, in proportionto their chiral angle. The thermodynamic nucleation barrier,on solid catalyst, is lower for the kinkless edges of achiraltubes. In spite of complex and random variability of numer-ous atomic structures in the process, the overall product abun-dance can be summed up in a remarkably compact mathemat-ical expression: x e − β · x . For lower temperatures and solid cat-alyst this function has a sharp maximum near zero, which ex-plains the observations of near-armchair nanotubes in exper-iments. Higher T and liquid catalyst make contact energiesrelatively equal ( β →
0) and nucleation of various types sim-ilarly probable, with the abundance then nearly proportionalto chiral angle. This demonstrates that the approach is suffi-ciently comprehensive, being able to explain rather disparatefacts accumulated over decades of experiments, from broaderchiral distributions to very narrow, almost single type peaks.Furthermore, we believe that the gained new insight must en-able finding ways to engineer chiral-selective nanotube pro-duction, thus advancing variety of long-awaited applications,all pending availability of properly pure material.
Acknowledgments:
Computer resources were providedby National Energy Research Scientific Computing Cen-ter, which is supported by the Office of Science of theU.S. Department of Energy under Contract No. DE-AC02-05CH11231; XSEDE, which is supported by NSF grantOCI-1053575, under allocation TG-DMR100029; and theDAVinCI cluster acquired with funds from NSF grant OCI-0959097. ∗ These authors contributed equally. † Correspondence to: [email protected][1] Iijima, S. Helical microtubules of graphitic carbon.
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