Systematic Enumeration of sp3 Nanothreads - Nano Letters (ACS

Letter pubs.acs.org/NanoLett

Systematic Enumeration of sp3 Nanothreads En-shi Xu,*,† Paul E. Lammert,*,† and Vincent H. Crespi*,†,‡,§,∥ †

Department of Physics, ‡Department of Materials Science and Engineering, §Department of Chemistry, and ∥Materials Research Institute, Pennsylvania State University, University Park, Pennsylvania 16802, United States S Supporting Information *

ABSTRACT: Slow decompression of crystalline benzene in largevolume high-pressure cells has recently achieved synthesis of a novel one-dimensional allotrope of sp3 carbon in which stacked columns of benzene molecules rehybridize into an ordered crystal of nanothreads. The progenitor benzene molecules function as sixvalent one-dimensional superatoms with multiple binding sites. Here we enumerate their hexavalent bonding geometries, recognizing that the repeat unit of interatomic connectivity (“topological unit cell”) need not coincide with the crystallographic unit cell, and identify the most energetically favorable cases. A topological unit cell of one or two benzene rings with at least two bonds interconnecting each adjacent pair of rings, accommodates 50 topologically distinct nanothreads, 15 of which are within 80 meV/carbon atom of the most stable member. Optimization of aperiodic helicity reveals the most stable structures to be chiral. We generalize Euler’s rules for ring counting to cover this new form of very thin one-dimensional carbon, calculated their physical properties, and propose a naming convention that can be generalized to handle nanothreads formed from other progenitor molecules. KEYWORDS: Nanothread, topology enumeration, sp3 carbon, benzene, DFT

C

down the column. This methodology can assist in (i) determining the experimental structure of current nanothreads, (ii) modifying reaction kinetics to produce new threads, and (iii) guiding theoretical calculations of nanothread properties, such as notable investigations into their high strength26 and their identification through NMR chemical shifts.27 The second point is especially intriguing, since kinetic control over the way that aromatic precursors bond together along one-dimensional columns, possibly obtained through chemical substitutions that induce differential reactivity, variations in progenitor crystal structure, or control of reaction conditions of temperature, pressure, and possibly photochemistry, could enable synthesis of multiple nanothread structures. As an initial demonstration of the method, we calculate the structural properties of the 15 lowest-energy pure-carbon nanothreads. Topological Enumeration. Each benzene precursor has six sp2 atoms that will convert to sp3 in the nanothread, thus it can be thought of as a superatom with a valence of six. Some of these six bonds will project up the crystallographic column that defines the nanothread axis, and some will project down. Assign a precursor ring to class I, II, or III, respectively, according to whether the partitioning is 1|5, 2|4 or 3|3. Because different classes cannot be mixed within the same column, an entire nanothread is also assigned a class accordingly. The severely unbalanced bond distribution in class I results in many four-fold rings and a correspondingly large energy penalty; we therefore

arbon chemistry is profoundly enriched by the extraordinary kinetic stability of carbon covalent bonds, ranging from sp2-bonded fullerenes,1,2 nanotubes,3,4 graphene,5,6 and graphite7 to sp3-bonded adamantanes (sometimes polymerized8,9), graphane,10−12 graphene fluoride,13−19 and diamond carbon in bulk, thin film, and nanowire forms.20 Very recently, a new route to the synthesis of one-dimensional sp3 carbon with the outer surface terminated in hydrogen, was discovered: topochemical constraints within compressed benzene crystals. When such crystals are decompressed slowly, the usual disordered reaction product is avoided and ordered arrays of thin one-dimensional nanothreads form as each carbon atom within a stack of benzene molecules bonds with a carbon atom in the molecule above or below.21 The end result is an ordered two-dimensional array of one-dimensional strands of sp3 carbon about as wide as the progenitor benzene molecules, called nanothreads. Experimentally, the precise atomic structure of carbon nanothreads is not yet known. Theorists made one-off predictions of a nanothread on at least three occasions: in 2001 by topological analogy to very narrow sp2 carbon nanotubes,22 in 2011 through direct simulation of high-pressure benzene,23 and in 2013 by extension of twistane molecules into an extended helix.24,25 These theorists predicted three different nanothread structures, underlining the question of just how many geometries may be possible and how to rationalize and navigate the possibilities. Here we present a systematic topological and structural enumeration of all possible bonding geometries within a one-dimensional stack of six-fold rings, each ring having six covalent bonds to neighboring rings up and © 2015 American Chemical Society

Received: April 7, 2015 Revised: July 10, 2015 Published: July 24, 2015 5124

DOI: 10.1021/acs.nanolett.5b01343 Nano Lett. 2015, 15, 5124−5130

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The double and quadruple interbenzene bondings within class-II nanothreads are treated similarly with cyclic permutations. We emphasize that for double connections, permutation means exchange, and both possibilities (a,b) → (c,d) and (a,b) → (d,c) are allowed. But there is no necessity that either arrangement actually has crossed bonds in a relaxed structure as a relative twist between the rings will uncross them. Figure 2 shows three examples of this sort of construction.

exclude class I from further consideration. (In any case, its analysis is much simpler than the other two classes.) A key concept here is that of topological unit cell, the fundamental repeat unit for the bonding pattern. This can differ from the standard crystallographic unit cell due to nontranslational symmetries (screw operations, reflections) or possibly symmetry-breaking conformational changes. For simplicity, we consider structures with at most two benzene rings per topological unit cell. Complete specification of the bonding pattern of a nanothread requires not only the class but knowledge of which particular atoms on a ring bond up, which down, and to which atoms on the neighboring rings they are bonded. The first refinement is handled by connector patterns as shown in Figure 1. These give the relative positions of atoms all bonding

Figure 1. Connector patterns of class-II and class-III benzene-derived nanothreads mark which atoms in a given benzene ring bond up or down the column of stacked rings.

in the same direction along the nanothread axis and are denoted according to an obvious nomenclature: (1,2), (1,3), and (1,4) for class II (with 4-prong partners not shown); and (1,2,3), (1,2,4), (1,2,5), and (1,3,5) for class III. Given an upconnector pattern on ring 0 and a compatible down-connector pattern on ring 1, the elements are bonded together in pairs. Here we impose a chemically motivated constraint to avoid crossed covalent bonds. Walking clockwise through the sites of the up-connector pattern of ring 0, their bonding targets on ring 1 must also cycle clockwise. We illustrate the considerations for class III. 1. Choose an up-connector pattern for benzene ring 0, say (1,2,3). 2. Choose an up-connector pattern for ring 1 (the next up the column of the nanothread), say (1,3,5). This leaves (2,4,6) as down-connector pattern. 3. Cyclically permute the down-tuple on ring 1 to obtain the possible bondings between rings 0 and 1:

Figure 2. Topological schematic and relaxed structures of three nanothreads, labeling atoms in each progenitor benzene ring to show the interconnection patterns. The polymer I phase of compressed benzene23 (top) is the lowest energy class-II nanothread. Polytwistane24 (middle) belongs to class III and is the lowest energy nanothread overall; both rings in polytwistane have the same connector pattern. The lowest energy soft chiral nanothread (bottom) has different connector patterns between the two pairs of rings.

Because class III has 4 connector patterns, there are 10 combinations of constituents for a two-ring cell and, naively, 3 × 2 = 6 ways to bond them. This gives an upper bound of 60 distinct class III nanothreads. A more careful count is presented in the Supporting Information, where it is shown that there are 10 one-ring-unit-cell and 36 two-ring-unit-cell class III nanothreads. Among those are 4 one-ring and 16 two-ring enantiomer pairs, so that up to inversion the numbers of distinct class III nanothreads with one and two rings per topological cell are 6 and 20, respectively. Similarly, there are 36 class II nanothreads with a two-ring topological cell, among which are 12 enantiomer pairs. Class II nanothreads cannot have a one-ring topological unit cell. There is a modest overcount here because this classification scheme dignifies the

(1, 2, 3) → (2, 4, 6) (1, 2, 3) → (4, 6, 2) (1, 2, 3) → (6, 2, 4)

Note that this conforms with the restriction mentioned earlier and that we could equally cyclically permute the ring 0 tuple but doing both would be redundant. 4. Repeat the previous step with the up-connector pattern of ring 1 and the down-connector pattern of ring 2, which is the same as that of ring 0. 5125


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Nano Letters Table 1. Nanothread Properties: Materials Properties of the Lowest-Energy Nanothreadsa Topology

identifier

Helical Interpolation

ring count (n4, n5, n6, n7, n8, n10)

energy per (CH)6 (eV)

Young’s modulus (free, pinned) (TPa)

Periodic approximant

λ atoms/Å

screw (trans., rot.) (Å, °)

Reff (Å)

Egap (eV)

nC

lC−C (Å)

1.43 1.40 1.40 1.41 1.69 1.35

3.89 4.79 4.51 4.55 4.48 4.11

1 4 4 3 4 4

1.54··1.57 1.53··1.60 1.53··1.59 1.54··1.58 1.53··1.59 1.51··1.67

Achiral 123456a 135462b 143562 135462 153624 143562

(0, (0, (1, (0, (0, (0,

0, 4, 2, 4, 4, 2,

6, 0, 2, 0, 1, 2,

0, 0, 0, 0, 0, 2,

0, 2, 0, 2, 0, 0,

0)* 0)* 1)* 0)* 1)* 0)*

0.73 0.82 0.95 0.97 1.01 1.04

1.16 0.98 0.93 0.90 0.59 1.08

143652c 136254 136425 135462

(0, (0, (0, (0,

0, 2, 2, 4,

6, 2, 3, 0,

0, 2, 0, 0,

0, 0, 1, 2,

0)* 0)* 0)* 0)*

0.57 0.62 0.70 0.81

(1.11, (0.73, (0.64, (0.63,

135246 132546 134562 145263 136524

(0, (0, (0, (0, (0,

4, 2, 2, 2, 2,

0, 2, 2, 2, 2,

0, 2, 2, 2, 2,

2, 0, 0, 0, 0,

0)* 0)* 0)* 0)* 0)*

0.64 0.66 0.69 0.75 0.96

Stiff, Chiral 1.14) 0.74) 0.64) 0.76) Soft, Chiral (0.31, 0.37) (0.35, 0.37) (0.08, 0.10) (0.19, 0.26) (0.41, 0.45)

2.79 2.41 2.38 2.60 2.60 2.44

4.30 4.98 5.04 9.23 9.22 4.91

2.45 2.75 2.63 2.64

(0.82, (4.37, (4.57, (2.27,

130.0) 160.0) 164.7) 134.8)

1.29 1.97 1.88 1.58

3.52 4.27 4.28 4.55

1 12 12 6

1.54··1.57 1.53··1.58 1.53··1.57 1.54··1.57

2.66 2.72 2.91 2.74 2.38

(4.51, (4.42, (4.13, (4.39, (5.05,

115.3) 079.2) 039.7) 102.9) 086.3)

2.31 2.10 4.09 2.44 2.26

4.23 4.16 4.53 4.19 4.24

12 12 12 12 12

1.53··1.58 1.53··1.58 1.53··1.58 1.53··1.58 1.54··1.59

a The ring count is restricted to the topological unit cell. Young’s moduli of chiral nanothreads are given for two different boundary conditions: free ends that can rotate in response to extensional torsion and pinned ends that cannot. Total energies are relative to a single sheet of graphane. The linear carbon atom density λ is used to define an effective area for the calculation of Young’s moduli. The effective radius Reff, band gap Egap (DFT/ PBE), number of symmetry inequivalent carbon atoms nC, and C−C bond length range lC−C of chiral threads are calculated for periodic approximants. Previously studied nanothreads include (a) tube (3,0), (b) polymer I, and (c) polytwistane.

energy soft chiral nanothread. The initial “1” in each name is universally present and hence strictly unnecessary, but we include it for clarity. The sp2-bonded fullerenes have a classic topological rule governing the count of rings, namely ∑i (i − 6)Fi = 12(g − 1), where Fi is the number of rings (faces) with i atoms and g is the genus (number of “holes” in the structure). g = 0 for a closed fullerene, 1 for a nanotube (notionally, it can be wrapped into a torus), and g ≥ 2 for various nanoporous Schwartzites.28 The rule derives from the formula Vertices − Edges + Faces = 2 − 2g for the so-called Euler characteristic of a graph in the special case where each vertex (atom) is connected by edges (bonds) to exactly three neighboring vertices, that is, sp2 coordination. This rule orients our thinking about the global structural topology of sp2 carbon.2 The vertices of sp3 nanothreads are also three-fold coordinated to neighbor carbon atoms and the nanothread genus is 1 as for a nanotube (again, thought of as a torus). However, unlike a fullerene, a nanothread can be so thin, held together by just two or three axial bonds, that identification of the rings (faces) may seem problematic. The problem is surmountable, however, as the Euler formula really applies to a so-called cellular embedding of a graph in a surface. An “embedding” is a drawing of the graph on a surface without crossing of edges, and “cellularity” simply means that the faces, that is, the pieces into which this surface is cut by slicing along the edges, do not have any interior holes (in mathematical terms, each face is “simply connected”). The chemically reasonable requirement that none of the polymerization bonds between benzene rings cross guarantees that its bonding graph can be cellularly embedded in a cylinder (really a torus when we take periodicity into account) by a simple projection: all of the faces project outward from the nanothread onto an imaginary surrounding cylinder without any of the projected

inter-ring C−C bonds with a special role in structural classification, whereas the carbon−carbon bonds in a completed nanothread cannot always be unambiguously disentangled into intra- and inter-ring bonds: certain completed thread structures can be notionally “sliced” into stacks of sixfold rings in more than one way with different slicings in different classes. Thus, more precisely, these 6 + 20 + 24 = 50 distinct thread connectivities can be thought of as 50 distinct polymerization pathways rather than 50 distinct structural outcomes. If it were to become an issue, the class-crossing redundancy in naming could be lifted by eventual knowledge of kinetic pathways to polymerization. A nanothread can be given a unique, parsimonious identifier by specifying the upward and downward bonding topology of a single ring. Take the polymer I structure of Wen et al.,23 shown at the top of Figure 2, as an example of class II. Number the atoms of the upper ring (red in Figure 2) so that its atom #1 connects to atom #1 of the lower ring (blue). Atoms (1,5) of the upper ring connect to atoms (1,3) of the lower ring. Meanwhile, atoms (2,3,4,6) of the upper ring connect to atoms (6,2,4,5) of the ring above it, because the third ring repeats whatever pattern is on the first ring (remember that we restrict ourselves to systems with two benzene rings per topological unit cell). Now we condense the nanothread’s name by focusing on just the middle ring. We write an ordered sequence of six integers, some underlined. The number in the first position represents the bonding target for the first atom in the middle ring: if underlined, the target is in the ring below; otherwise it is in the ring above. And similarly for the second through sixth integers in the sequence. The upper example in Figure 2 shows how this works for polymer I, which is called (135462). The other two examples shown are the nanothread with the colloquial name “polytwistane”,24 and the lowest 5126


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optimize the structure of each helical nanothread, we perform a systematic series of first-principles calculations in periodic unit cells with different screw angles (the rotation angle of one helical repeat unit) and screw translations (the axial length of one helical repeat unit). We vary the screw angle over a range of about ±20° by changing the number of helical repeat units within a cell of 2π, 4π, or 6π (whichever is computationally practical) at each of a series of screw translation distances, thereafter interpolating the screw angle quadratically and the screw translation according to the Birch-Murnaghan equation of state to obtain an optimal aperiodic helix. Figure 3 and Table 1 report the optimal screw operations calculated in this way. Interpolation is straightforward for the total energy but is potentially more problematic for other material properties such as bandgaps that might not be smooth linear functions of the screw operation due to band crossings and so forth. For such properties, we use a periodic approximant, whose total energy is typically within about 0.02 eV per (CH)6 formula unit of the ideal interpolation. Similar issues of aperiodicity also arise for chiral sp2 nanotubes.43 Figure 3 shows the 15 most stable nanothreads, along with their energies per (CH)6 formula unit relative to an isolated sheet of graphane. Three nanothreads previously proposed in the literature, the (3,0) sp3 nanotube,22 polytwistane,24 and the polymer I phase of compressed benzene,23 all appear in the enumeration as highly stable with chiral polytwistane the lowest energy allotrope overall and the (3,0) sp3 nanotube and polymer I the two lowest energy achiral nanothreads. Many structures reside within a few tens of millielectronvolts per carbon atom of each other, so kinetic pathways will likely determine the actual nanothread(s) formed in experiment. Structures 135462 and 153624 have crystallographic unit cells with 24 carbon atoms, twice the topological unit cell. They accommodate this through 90° and 180° screw rotations, respectively. The chiral threads have long translational unit cells but only 2 carbon atoms (for 143652), 6 carbon atoms (for 135462), or 12 carbon atoms in the helical repeat unit. Polytwistane has the structure of an externally hydrogenated (2,1) nanotube with a screw angle close to that arising from a purely geometrical construction of a helical unit cell from a hexagonal network. The highest-energy nanothreads (not presented here) have many four-fold rings; this includes the class I nanothreads, which are all at least 1.67 eV per (CH)6 higher in energy than graphane. The lowest-energy nanothreads tend to be chiral; it may be that the extra long-wavelength structural degree of freedom implicit in the helicity allows these threads to better optimize their structural coordinates. Overall, nanothread binding energies are well within the range of known CxHx hydrocarbons. Comparison to pure-carbon sp2 systems requires a reference state for the excess hydrogen. Choosing molecular hydrogen for this, the lowest-energy nanothread is 0.68, 0.90, and 0.98 eV per six carbons lower in energy than graphene, a (10,10) carbon nanotube, and a low-energy carbon allotrope consisting of pentagons and heptagons.44 Bond stretching (C−C) and bending (C−C−C) energetics can be captured in approximate form by calculating meansquared deviations

edges (bonds) crossing each other. (Imagine a light source along the nanothread axis, with the cylinder as a projection screen. The shadows of the bonds do not cross.) To perform the projection, thread a centerline (our imaginary light source) through all the progenitor benzene rings in the stacked-ring representation on the left side of Figure 2. These rings run around the circumference of the cylinder and thus do not appear as faces in the Euler analysis. If a class-II nanothread has crossed bonds (as described earlier), then the centerline must also be threaded through the ring that contains the crossed bonds, so that the crossing does not project outward onto the cylinder. It is possible for the centerline to puncture some additional rings: if a uniform sense of circulation is applied to each progenitor benzene molecule, then any interbenzene ring on which can be defined a sense of circulation consistent with both progenitors is also punctured. All these punctured rings are excluded from Euler’s rule. The remaining rings enter Euler’s rule, as for g = 1 fullerenes: V − E + F = 0. With two benzene rings per topological unit cell, we have V = 12; because 3V = 2E, we conclude that there are always six faces (i.e., rings) within the topological unit cell, and those below six-fold are compensated by those above just as for an sp2 nanotube: ∑i (i − 6)Fi = 0. Table 1 provides detailed ring statistics. The punctured rings, which must be excluded from Euler’s rule, are nevertheless still structurally relevant. Fortunately, in many cases the punctured rings themselves satisfy ∑i (i − 6)Fi = 0. Using the ring definition that King29 developed for amorphous systems, we find that 11 of the 15 lowest-energy nanothreads satisfy this broader interpretation of Euler’s rule; these are marked with an asterisk in Table 1. Other ring definitions used in the analysis of amorphous solids30−32 or so-called essential rings33 are too aggressive for our purposes, in that they exclude all rings above six-fold. Current experimental results appear consistent with at most only occasional cross-links between adjacent nanothreads within the sample.21 To the extent that such cross-links are produced through the loss of hydrogen they will not modify the Euler analysis given above. Nanothread Properties. Having laid out a methodology to systematically generate nanothreads, we now calculate their physical properties. The topological repeat unit constructed above does not necessarily coincide with the crystalline unit cell. So as not to presuppose the actual crystalline unit cell, the topologically defined structures for all possible nanothreads with up to two benzene units per topological repeat unit were initially relaxed as finite-length molecules containing at least four topological unit cells (≥8 benzene rings) with free boundary conditions, using a bond topology preserving potential, MMFF94.34,35 The rings at either end of the relaxed thread were then discarded and a proper crystalline unit cell (or helical cell) was extracted from the interior, to build an axially periodic system for further relaxation and study at a density functional level. Density functional calculations used the generalized gradient approximation (PBE) as implemented in VASP36−42 with a 600 eV energy cutoff, a 0.01 eV Gaussian smearing, and a line of kpoints along the axial direction separated by less than 0.24 Å−1. These parameters ensure a convergence of 0.01 eV per (CH)6 formula unit (about 1 meV per atom). For the achiral structures, the axial unit cell length was fit to a Birch− Murnaghan equation of state and the internal coordinates were relaxed again at the optimal lattice constant. The lowest-energy chiral threads found are helical, presumably with an aperiodic ideal screw angle. To properly

δl 2 =

∑

(l − l0)2 , N

δθ 2 =

∑

(θ − θ0)2 N

By fitting generalized bond stretching and bending stiffness parameters and also l0 and θ0, we obtain a quadratic form 5127


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Figure 4. An empirical formula for the total energy of the nanothread in terms of just the root mean squared deviations in bond length and angle, together with a reference energy, gives a reasonable description for the trends in the structural energetics of the lowest energy nanothreads.

λ, similar to the method used in Dragan et al.22 and systematized by Arias.45 Multiplying λ by a reference volume V0, such as the volume per carbon atom in bulk diamond, yields an area λV0 that takes the place of cross-sectional area in the definition of Young’s modulus. Helical nanothreads couple axial extension to torsion at first order; those with freely rotating ends are less stiff. We report both Yfree and Yfixed, the first being extracted from the helical interpolation described earlier. A nanothread’s extensional stiffness is largely determined by the degree to which axial extension can be accommodated by bond bending versus bond stretching. The long-period “open” helices in the lower third of Figure 3 (135246, 132546, 134562, 145263, and 136524) suffer the worst from this effect, as does the alternating ladder structure of 153624. The openness of the helix can be characterized by the maximum radial deviation of a carbon atom from the screw axis. As shown in Table 1, the softer chiral nanothreads have wider helices. The alternating ladder structure of 153624 is also significantly wider and softer than its achiral peers. Elastic nonlinearities should stiffen these softer threads under greater extension, because extension through bond bending becomes less effective as the threads straighten out. The stiffest nanothreads tend to have a small number of symmetry-inequivalent atoms and thus fewer prospects for anomalous weak points in the structure, for example, polytwistane (143652) and the (3,0) sp3 tube (123456). These better-converged calculations give a somewhat smaller Young’s modulus for the (3,0) sp3 tube than was reported in its original description.22 We performed our calculation on ideal periodic systems but in reality the nanothreads could have more complex, potentially defective structures. An analysis within the classical ReaxFF force field46 yields reasonable agreement for the extensional stiffness of the ideal (3,0) sp3 tube (1.27 TPa); a prior analysis using the same force field reveals that the introduction of defects into the nanothread tends to soften the system.26 For comparison, the

Figure 3. Lowest-energy nanothreads displayed as the DFT-relaxed structures with the energy per (CH)6 formula unit relative to graphane. Helical threads also show translations and rotations for the fundamental screw operation. Previously studied nanothreads are (a) sp3 tube (3,0), (b) polymer I, and (c) polytwistane.

Eempirical ≈ E0 + κstrδl2 + κbendδθ2 that is plotted in Figure 4 against first-principles results (treating chiral nanothreads as periodic approximants), showing reasonable agreement. The range of bond distances observed is typical for saturated hydrocarbons. The unusually long 1.67 Å bond in nanothread 143562 reflects internal tension associated with the periodic boundary conditions having straightened out a structure that prefers to curl up. Because cross-sectional area is ill-defined at the nanoscale, we quantify the extensional stiffness in terms of linear atom density 5128


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carbyne polymer is significantly stiffer under extension (3.46 TPa, using our definition of cross-sectional area)47 because bonding stretching is the only choice for accommodating extension. The bandgaps of the nanothreads are typical of saturated hydrocarbons (remembering that these DFT-level gaps within the generalized gradient approximation underestimate the true gap). Interestingly, the most stable achiral and chiral threads also have the smallest bandgaps, which contradicts the structural intuition that more stable structures tend to have larger gaps. Higher-level quasiparticle calculations may resolve this discrepancy. Although the gaps of pure-hydrocarbon nanothreads are uniformly large, substitution of nitrogen for CH as in pyridine or pyrimidine48 provides a possible route toward obtaining structures more amenable to doping and charge transport. Substitution of hydrogen in the precursor rings with electron donating or withdrawing groups could modulate the reactivity of the ring in a site-specific manner, giving some measure of control over which nanothread connection pattern occurs. However, any change to the molecular geometry of the precursor will also affect its crystal lattice and thus the topochemical constraints. Fortunately, firstprinciples methods of crystal structure prediction are well advanced and thus will provide some guidance on the geometry of the topochemical constraint, as knowledge of reaction kinetics improves. Larger aromatic ring systems provide another means of designing desirable properties into nanothreads, including the extension of the hydrogenated nanotube analogy to slightly larger systems such as (3,1) and (2,2). (The (2,2) thread can also be constructed from a unit cell with three benzene rings). A fully rationalized design methodology awaits deeper understanding of the reaction kinetics, a topic of great interest although one outside our immediate focus on defining and systematizing the landscape of possibilities.

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REFERENCES

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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.5b01343. Group theoretic analysis for reducing enumeration reduncancy. (PDF) Atomic coordinates of the reported nanothreads in .cif formats. (PDF)

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Letter

AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported as part of the Energy Frontier Research in Extreme Environments (EFree) Center, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science under Award Number DESC0001057. The simulation is supported in part by the Penn State Materials Computation Center. The authors thank Roald Hoffmann and John Badding for valuable discussions. 5129


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Systematic Enumeration of sp3 Nanothreads - Nano Letters (ACS

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