Transient Kinetic Selectivity in Nanotubes Growth on Solid Co–W

Letter Cite This: Nano Lett. XXXX, XXX, XXX−XXX

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Transient Kinetic Selectivity in Nanotubes Growth on Solid Co−W Catalyst Evgeni S. Penev,† Ksenia V. Bets,† Nitant Gupta, and Boris I. Yakobson* Department of Materials Science and NanoEngineering, Rice University, Houston, Texas 77005, United States

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S Supporting Information *

ABSTRACT: Solid Co−W catalysts have been shown to yield single-walled carbon nanotubes (CNT) with high selectivity, simplistically attributed to CNT-catalyst symmetry match for certain chiral indices (n,m). Here, based on large-scale first-principles calculations combined with kinetic Monte Carlo simulations, we show instead that such selectivity arises from a complex kinetics of growth. The solid Co7W6 catalyst strongly favors a restructured, asymmetric CNT edge which entails preferential nucleation of tubes with 2m < n but much faster growth of chiral tubes with n ⩽ 2m. We uncover a tendency of interface defects formation that, although rare, drive CNT type change from smaller to larger chiral angles (zigzag to armchair). Being both least prone to defects and fast growing, the (12,6) CNT appears as a transient, kineticsselected type reaching highest abundance. KEYWORDS: Carbon nanotubes, chiral selectivity, solid catalyst, growth kinetics, atomistic modeling χ-sensitive, leading to a broader abundance distribution, increasing toward armchair types.10 Such kinetics route of chiral selectivity, controlled by the 9 factor in eq 1, may even yield preferentially the fastest growing midchiral (2m,m) tubes, with χ = tan−1(√3/5) ≃ 19°, where the number of kinks at the edge reaches maximum m.11 Naturally, the sharp peak can be mitigated by roughness of the catalyst plane and configurational entropy. The promise of designed solid catalysts for achieving chiral selectivity was highlighted by recent experiments reporting high abundance of essentially single CNT type: (12,6) using Co7W612 or Mo2C,13 and (8,4) using WC catalyst.13 Both studies have attributed such selectivity to the structural templating (via “epitaxial matching”14) by specific crystal planes, resulting in nucleation of only (2m,m) CNT. Yet, such a rationale in view of eq 1 may seem puzzling: good matching means typically low interface energy, stronger bonding. Growth via such an interface, however, would be kinetically suppressed as carbon insertion would disrupt the tight, matching CNT−catalyst contact, resulting in high kinetic barrier and slow carbon accretion. Theoretically,5 one would expect abundance of “one-index-off” tubes, for example, (2m,m ± 1) or (2m ± 1,m ∓ 1), as a trade-off between energy-favored matching and the kinetic need for loose, accessible contact. On the basis of large-scale density-functional theory (DFT) energy and barriers calculations, we look for mechanistic understanding of the observed selectivity near (12,6) tubes on solid Co7W6. In the course of our study, it is revealed that

arbon nanotubes (CNTs) may be aptly called “topologically nontrivial” graphene, rolled up along a given direction C = na1 + ma2 on its honeycomb lattice, changing from genus-0 plane into a genus-1 hollow tube. This brings about a new descriptor of the lattice topology, aside of the a diameter d ≡ ∥ C ∥/π = π (n2 + nm + m2)1/2 − its helicity, or chirality, determined by the pair of non-negative integers (n,m), or conveniently described by the (“chiral”) angle χ ≡ tan−1(√3m/(2n + m)) between C and the zigzag motif of the lattice. Because all CNT properties are determined by the helicity, controlled synthesis of CNTs with defined (n,m) has been the tantalizing goal of catalytic chemical vapor deposition (CVD), the most promising way of synthesis. Yet, the vast parameter space (growth conditions and duration, carbon source, catalysts, and so forth) inherent to CVD has made the chiral-selective growth a vexing problem for both theory1,2 and experiment.3,4 A recent step in understanding why CNTs grow chiral5 was motivated by puzzling abundance of near-armchair (n,n − 1) tubes, observed repeatedly over the years.6−8 A key ansatz in ref 5 has been the factorization of the CNT-type abundance ( n , m into a product of a nucleation probability term 5n , m and a growth rate 9 n , m

C

( n , m = 5n , m·9 n , m

(1)

An analysis shows that on a generic solid particle, the convolution of interface energetics9 and kinetics10 leads to a peaked distribution near armchair (A) or zigzag (Z) types, ((χ ) ∼ χe−χ , taking behaviors in vicinity of χ-range ends as 5n , m ∼ e−χ and 9 n , m ∼ χ . On the other hand, on a liquid, fully compliant catalyst particle the nucleation term is only weakly © XXXX American Chemical Society

Received: June 5, 2018 Revised: July 2, 2018 Published: July 6, 2018 A

DOI: 10.1021/acs.nanolett.8b02283 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 1. (a) Hexagonal bulk unit cell of the trigonal μ-phase (space group R3̅m) of the Co7W6 intermetallic alloy16 (see also SI, Figure S1). (b) A (12,6) tube on a W-terminated (003) facet, also highlighted in (a), of a Co7W6 particle of diameter ≃2 nm. (c) The set of tubes of nearly the same diameter d ≃ 12.5 ± 0.2 Å (d variation is indicated by dashed arcs), studied here, shown on the (n,m) map.

Figure 2. (a) Total interface energy Γ(χ) of the tubes in Figure 1c on the W-terminated Co7W6(003) surface, for both conventional and A| Z-segregated edges. The top-right inset illustrates the A|Z decomposition of the conventional-edge chiral vector C. The images show the atomic configuration of the two types of edges for the example of the (11,7) tube. (b) Unfolded view of the relaxed CNTCo7W6(003) interface for the (11,7) tube with an A|Z-segregated edge. Nearest-neighbor metal atoms (only Co in this case) to the CNT edge are shown as bigger spheres.

CNT edge in contact with Co−W catalysts can undergo an armchair-zigzag “phase segregation” toward asymmetric edges15 with energies reduced and their χ-dependence largely suppressed, compared to the conventional straight-circular CNT edges. Such segregation substantially alters the carbon insertion order and kinetics and the overall edge growth mode. This can be realistically tracked by kinetic Monte Carlo simulations that allow us to uncover a growth scenario resulting in dominant (12,6) tubes. We begin by quantifying the nucleation (5 ) and growthrate (9 ) factors in eq 1 for a set of eight (n,m) CNTs of similar diameter on the (003) surface of the Co7W6 catalyst, Figure 1. The choice of surface and specific atomic termination (out of multiple possible) is partly based on the structural characterization and catalyst synthesis conditions in ref 12 (see the Supporting Information (SI), Figures S2−S4). Complete details of the computational setup and discussion of the methodological aspects are given in the SI (Figures S5 and S7). It has been conventional to consider the CNT edge being circular, corresponding to the chiral vector C, comprising 2m A atoms and (n − m) Z atoms (top insets, Figure 2). Among all tubes in the set, Figure 1c, the (12,6) interface has the highest (6-fold) rotational symmetry, where the A edge atoms are in contact with Co atoms, and the Z edge atoms have matching surface W atoms. Yet, symmetry match is present also for the achiral tubes. In fact, one can easily see that the (18,0) tube with slightly larger d would have a perfect one-to-one contact between all edge C atoms and substrate Co atoms (see SI, Figure S8). Moreover, although the symmetry matching appeals to intuition it must be judged by the interface energy value, which may be either minimum or maximum. The total interface energy, Γ(χ,d) = λ·γ(χ), where λ is an edge-length factor, is plotted in Figure 2. For the conventional edge with λ = πd, Γ shows the expected arch-behavior5 with local minima at the Z and A ends. Here, Z is a slightly favored interface, ΔΓ = ΓZ − ΓA < 0, qualitatively similar to a sphericaljellium catalyst model17 (an example of the opposite case, ΔΓ > 0, occurs for CNTs on the (111) surfaces of Ni and Co,5,18 with larger |ΔΓ|). This implies a vanishing 5 ∼ exp( −Γ/kBT )

and thereby suppressed nucleation of all but Z tubes under typical growth conditions. Such an observation compels us to address a more general problem. Within an equilibrium thermodynamic description of CNT nucleation on a solid catalyst, an interface energy with pronounced minima only for achiral tubes is difficult to reconcile with experimental evidence of selectivity for chiral tubes. Scenarios which do not involve such behavior include, for example, liquid catalysts,5 or eventually gradual change of CNT chirality.19−21 On the basis of CNT-catalyst interface thermodynamics,9 we find that Γ may be significantly lower if the CNT edge segregates into two contiguous Z- and A-domains (denoted by “A|Z”), Figure 2. The decomposition of the conventional edge is represented by the shaded triangle with base C (inset, Figure 2) and has two simply geometric consequences. First, the A|Zsegregated edge is longer, with λ = 2πd[sin χ + sin(30° − χ)], the length increases by a factor of 2 (2 − 3 ) cos(χ−15°) at most ≃3.5% in the midrange of χ. Second, the axis of a CNT in contact with a crystal plane of the catalysts is tilted ∼h/d off the plane normal. Note that for the achiral and minimally chiral (n,n − 1) or (n,1) tubes, there is no edge segregation. Figure 2 shows that the A|Z-segregated edge results in a significantly lower (∼1−3 eV on Co7W6) interface energy Γ for all chiral tubes. Note that this effect occurs due to the presence of solid catalyst, while the difference between the edge energies of the free conventional-straight and segregated edges is insignificant.18 The A|Z edge type separation essentially eliminates all but one kink, largely recovering the tight contact typical for the achiral interfaces, as evident from the unfolded map in Figure 2b for the (11,7) tube as an example. The large energy preference of the A|Z-segregated interface suggests that chiral CNTs will likely nucleate as caps B

DOI: 10.1021/acs.nanolett.8b02283 Nano Lett. XXXX, XXX, XXX−XXX

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Figure 3. Energetics ΔE of the carbon addition cycle (in terms of number of C2, N) for a growing A|Z-segregated interface. The range of each Naxis gives the cycle length Nc. (a) Familiar ΔE profile for the achiral tubes where addition of Nc = n dimers completes a new layer of hexagons. The dashed-line segment for (16,0) indicates states incorporating defects; (b−d) ΔE profiles for selected chiral tubes; (e−g) C2 addition sequence corresponding to the most probable pathway for the corresponding (n,m) (solid lines in the ΔE plots). Arrows indicate the directional tendency of the process.

all other types). Overall, the achiral tubes exhibit familiar ΔE profile,5 notably uneven for all chiralities. The growth patterns dictated by the energy profiles for the chiral tubes, Figure 3b− d, are shown in Figure 3e−g. Here, we simulate the growth process by evolving the system state using kinetic Monte Carlo (kMC)22−25 at T = 1030 °C, close to experiments12 (for details see the SI, Figures S9 and S10). A kMC trajectory initiates with the first dimer added to the available kink, forming two new kinks, one on the A-facet and another on the Z-facet. Effectively, new hexagon layers simultaneously nucleate on both the A and Z facets. We find that chiral CNTs close to A with n ⩽ 2m, like (10,8), (11,7), and (12,6), are fastest growing (see SI, Figure S13d) and that they elongate through addition of hexagon layers to the Z-facet (Figure 3e,f), whereas those closer to Z, like (13,5), (14,3), and (15,2), advance by kink-flow along the A-facet (Figure 3g), as indicated by the arrows. With growth rates determined from the kMC simulations, the overall mass distribution appears broader than the population abundance (nucleation trend) dictated by the interface energy in Figure 2 (see SI, Figure S13a,e). Yet, the zigzag (16,0) CNT in Figure 3 is representative of a pronounced tendency we find for tubes closer to Z, where interface defects can form that would change the CNT chirality, shifting it toward armchair. This is less likely to occur for tubes closer to A, n ⩽ 2m, and essentially improbable for the armchair (9,9) tube (see SI, Figure S14). In Figure 4, a sequence of events is illustrated that can lead to helicity switch (n,m) → (n′,m′): inclusion of a 5|7 dislocation defect whose Burgers vector orientation determines the new chiral indices (n′,m′).17,26,27 In Figure 3a, for example, a pentagon formation is favored at N = 11 and subsequently a 5|7 dislocation, resulting eventually in a chirality switch (16,0) → (16,1).

with such edge. Moreover, we have found that the conventional interface is kinetically unstable upon C2 dimers accretion18 and segregates into A|Z form. Once the CNT−catalyst interface undergoes the A|Zsegregation, Figure 2, the growth mechanism of chiral CNTs departs significantly from the screw-dislocation theory.5,10,11 In the latter, the growth occurs via conventional-edge interface, its rate is proportional to the number of available kinks, 9 ∝ m /d ∼ χ , and a straightforward analysis following ref 5 shows preference for minimally chiral CNTs (see SI, Figure S11). The clear relation 9 ∼ χ is no longer valid for the A|Zsegregated edge as it has a single kink between the Z and A “facets”. Because disrupting the tight contact at these two facets is rather energetically taxing, C2 insertion is initiated at the single kink and leads to fluctuating, small number of kinks all along, regardless of χ. Apparently, unlike the conventional chiral edge5,9,10 the A|Z-segregated edge cannot be trivially repeated upon C-dimer addition and we expect more complex energy profile characterizing the growth, where even the existence and nature of an accretion “cycle” is a priori unclear. The atomically rough Co7W6(003) surface, exposing both Co and W (the packing density of this W-termination is about four times smaller than that of the close-packed (111) surface of fcc Co), suggests that the minimally chiral (n,n − 1) and (n,1) tubes may also alter the simple one-dimer addition cycle.5,10 To quantify the CNT growth mechanism through an A|Zsegregated edge we compute the energy changes ΔE upon C2 addition (details in the SI). For each (n,m) we sample multiple possible, symmetry-inequivalent insertion sites/kinks along the interface. A CNT grows by cycling through a subset of local minima (data points in Figure 3a-d). The solid lines in Figure 3a−d show the ΔE(N) profiles of the most probable pathways for a few representative helicities (n,m) (see SI, Figure S6, for C

DOI: 10.1021/acs.nanolett.8b02283 Nano Lett. XXXX, XXX, XXX−XXX

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5b. This notable outcome appears somewhat surprising given the mere 512,6 ≃ 0.5% nucleation probability. Detailed analysis of the kMC trajectories, however, reveals a nontrivial growth scenario where the CNT population is largely determined by chirality switching events driven by defects formation, in departure from eq 1. We find that the observed probability of such events to occur during the simulation timespan is ∼1 for (16,0) and near-Z tubes and is lowest for (12,6) among all chiral tubes (see SI, Figure S14). The key to understanding such behavior is the presence of large-ΔE steps in the carbon addition cycle of chiral tubes, for example, insertion of the N = 7 carbon dimer for (13,5), Figure 3d, or the first dimer for (11,7), Figure 3c. In such transitions, the CNT edge is prone to forming dangling-C2 configurations, Figure 4, that enable formation of the chirality-switching 5|7 dislocations with their Burgers vector oriented so that the chiral angle χ is incremented. This mechanism is reflected in the broad L distribution for the (12,6) tube, Figure 5c. Although (12,6) barely nucleates, an initial rapid chirality switching of the tubes with χ < 19°, leads to a burst-like increase in (12,6) population (t ∼ 0−3 min), Figure 5d. A straightforward refinement of the CNTs population estimate [i.e., number of CNTs segments of specific (n,m)] is to allow for a length threshold, Lc, in CNT counting, below which a CNT segment cannot be “detected”. The population sensitivity to Lc is shown in Figure 5e. The distribution is practically unchanged for Lc ≳ 50 nm, and at this threshold the population abundance of (12,6) is ≃80%. Lastly, the overall mass abundance in Figure 5f displays essentially a single dominant peak for (12,6) with ∼90% selectivity, similar to experiment,12 and is not sensitive to Lc in the considered range where Lc ≪ ⟨L⟩.

Figure 4. Schematics of reversible carbon-dimer addition (±C2) to the A|Z-edge, leading to a 5|7-dislocation and a subsequent helicity switch (n,m) → (n′,m′), or recovery-healing via bond rotation.

By combining the A|Z-segregated interface energetics (Figure 2) with the ensemble of generated kMC trajectories over the ΔE(N) profiles (Figure 3) explicitly including configurations that lead to or contain topological defects (allowing also for their “healing”28,29), we uncover a model picture of CNT growth using the solid Co7W6 catalyst, compatible with experimentally reported abundance.12 The nucleation probability 5(χ ), Figure 5a, favors strongly the chiralities toward the Z end, n > 2m, from (16,0) to (13,5), reflecting the high stability (low Γ) of their A|Z-segregated interfaces. An initial sample of 105 tubes is then generated using the computed population distribution 5(χ ), Figure 5a. The growth of this sample is simulated by kMC trajectories of t = 900 s timespan with resultant mean length distribution ⟨L(χ)⟩ of CNT segments peaked at (12,6), as shown in Figure

Figure 5. (a) Nucleation probability 5(χ ) of CNTs with A|Z-segregated edge at T = 1030 °C. (b) Mean length ⟨L(χ)⟩ of a sample of 105 tubes with initial chirality distribution from (a), determined from t = 900 s kMC simulation. (c) Length L distribution for the (12,6) tube (obtained by using 0.1 μm bin size), the horizontal dotted line indicating the mean length. (d) Dynamics of the (12,6) chirality population due to (13,5) → (12,6) and (12,6) → (11,7) transitions over the kMC simulation time-span. (e) Population abundance as a function of length cutoff Lc, marked at each curve (in nm): only tubes with L ⩾ Lc are counted. The maximum plotted Lc = 100 nm corresponds to tubes 30−50 times shorter than the average tube for the sample. For clarity, each subsequent curve is vertically offset by 0.1. (f) Mass abundance of all CNTs (Lc ≡ 0), showing ∼90% selectivity for (12,6), only weakly sensitive to Lc. D

DOI: 10.1021/acs.nanolett.8b02283 Nano Lett. XXXX, XXX, XXX−XXX

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The Co7W6 catalyst and the more traditional Ni, Co, and Fe have shown disparate selectivity in single-walled CNT growth. Whereas the former is reported to exhibit essentially a monochiral preference for (2m,m)-tubes, the latter few produce notably near-armchair (n,n − 1) abundance. Such near-armchair preference, however, can be understood even within a continuum model of the CNT−catalyst interface.5 The present analysis shows that atomistic description is critical for understanding the true origin of the reported (12,6) preference. This can be attributed eventually to a number of distinct features of the Co7W6 catalyst. The bimetallic solid particles expose atomically rough facets, Figure 1b. The achiral CNT−Co7W6(003) interfaces are very similar in energy (|ΔΓ| ≃ 0.35 eV) and the chiral interfaces segregate into A and Z segments, Figure 2. This separation has a dramatic effect on both nucleation probability and kinetics of growth. It reduces greatly the interface energy variability among the chiral angles, makes these low-energy configurations prone to defect formation upon carbon insertion, and features just a single kink making growth still possible. Thus, even though CNTs with smaller chiral angle would be dominant in nucleation they undergo chirality switching as a result of defects incorporation. The (12,6) A|Z-segregated interface is found to be least prone to defects formation and appears as an “transient attractor” in the chirality evolution trend from Z to A. One can identify then growth conditions and time scale (see SI, Figure S15) over which this trend results in the overwhelming (12,6) mass abundance, Figure 5f. Although such remarkable selectivity can be explained within our formalism only by explicitly including defect formation mechanism, we do not see clear correlation with or necessity of structural or symmetry matching. The present study makes the understanding of (2m,m) preference using WC, Mo2C,13 and even NixFe1−x30 catalysts, yet more intriguing. So far, the recent work of Yang et al.12 appears unique: other attempts at using the Co−W intermetallic did result in (12,6) selectivity, albeit abundance is much lower and the catalysts phase is found to be a Co6W6C carbide.31 Nevertheless, the case of the solid Co7W6 catalyst demonstrates the ability of our formalism to uncover complex growth mechanisms and suggests its more general applicability to guiding the selection/design of catalysts for chiralityselective CNT synthesis.

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The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Science Foundation (CBET-1605848). Computer resources were provided by the National Energy Research Scientific Computing Center, supported by the DOE Office of Science under Contract No. DE-AC02-05CH11231; the DoD Supercomputing Resource Center; XSEDE, which is supported by NSF Grant OCI1053575 under allocation TG-DMR100029; and the DAVinCI cluster at Rice University, acquired with funds from NSF grant OCI-0959097.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.8b02283.

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REFERENCES

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The μ-phase of Co7W6: crystal structure, surfaces, and choice of surface termination; computational details: setup, definitions, benchmarks; kinetic modeling: barrier estimates, kMC protocol; CNT growth scenarios (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Evgeni S. Penev: 0000-0002-4783-8316 Nitant Gupta: 0000-0002-3770-5587 Author Contributions †

E.S.P. and K.V.B. contributed equally to this work. E

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