# Topological Anisotropy of Stone-Wales Waves in Graphenic Fragments

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## Abstract

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## 1. Introduction

_{q/r}(Figure 1a), associated to the most studied variants, the SW

_{6/6}in graphene (Figure 1b) often called Stone-Thrower-Wales rotation and the SW

_{5/6}in fullerenes (Figure 1c) the so-called pyracylene rearrangement.

^{2}-carbon systems recently appeared in literature (see [4] and related).

^{2}-hybridized carbon rings, detected by scanning tunneling microscopy (STM) images, function as a quasi-one-dimensional metallic wire and may be the building blocks for new all-carbon electronic devices. This important experimental finding enforces meanwhile the theoretical role of the SW waves, that are in principle structurally simpler than the pentagons-octagons chain reported in [6], as a possible hexagonal inter-grain spacing (see the visualizations given in Section 2) between graphenic fragments. Molecular mechanics simulations show that in graphene the presence of cylindrical curvature energetically facilitates such a split of the 5/7/7/5 SW dislocation dipole [4], assigning to this class wave-like atomic-scale rearrangements a fundamental role in nanoengineering of graphenic lattices. One has however to notice that other transmission electron microscopy (TEM) detailed measurements point out [7,8] that the migration and the separation of the pentagon-heptagon pairs does not happen on planar graphene membranes where the 5–7 defects relax back reconstructing the original graphene lattice. These experiments indicate that extended dislocation dipole, favored by the presence of structural strain, preferably appear in curved graphitic structures or systems like CNT or fullerene molecules. In epitaxial graphene grown at high temperatures on mechanically-polished SiC(0001), a characteristic 6-fold “flower” defect results from STM measures [9,10]. We note that the observed rotational grain boundaries is conveniently describable as radial type of the SW wave suggesting that the wave-like theoretical mechanism presented here, may have a general applicability.

_{p/r}varying the internal connectivity of four generic carbon rings made of p, q, r, s atoms to produce four new adjacent rings with p−1,q+1,r−1,s+1 atoms without changing the network of the surrounding lattice. SW

_{p/r}reversibly rotates the bond shared by the two rings p and r, preserving both, the total number of carbon atoms

_{6/6}rotation transforms four hexagons in two 5|7 adjacent pairs (Figure 1b) symbolized in literature [4,11] as 5/7/7/5 defect and also quoted as the SW defect or the dislocation dipole. We remember here that the SW rotations play an important role in connecting the isomers of a given C

_{n}fullerene with different symmetries. In the crucial case of the C

_{60}fullerene, its 1812 isomers are grouped by the pyracylene rearrangements SW

_{5/6}(Figure 1c) in 13 inequivalent sets (the larger one consisting of 1709 cages) connected to the buckminsterfullerene (C

_{60}-I

_{h}) through one or more SW transformations [12], leaving 31 isomers unconnected to any of these sets. This limitation has been overcome by the introduction of non-local generalized Stone-Wales transformations [13] to generate the whole C

_{60}isomeric space starting from just one C

_{60}isomer.

_{2v}symmetry; this barrier reaches 9 eV for hexagonal systems like nanotubes or large graphene portions. Using the extended Hückel method, enlarging the relaxation region around the SW defect, it can be found that the formation energy of a SW defect considerably decreases to 6.02 eV for a flat graphene fragment case. This result has been verified by using ab initio pseudopotential [14]. This result seems to preclude the formation of any SW 5/7/7/5 defect in nature, but as it has been reported [15,16] that this barrier drops rapidly, reducing to 2.29 eV the creation barrier of SW rotations due to the catalyzing action of interstitials defects or ad-atoms present in the hexagonal networks. Pentagon–heptagon pairs have been predicted to be stable defects also in important theoretical articles [17,18] showing that energetic particles, as electrons and ions, generate 5|7 pairs in graphite layers or CNT’s as a result of knock-on atom displacements. On the experimental side, accurate high-resolution TEM studies made on single-walled carbon nanotubes [19] or electron-irradiated pristine graphene [20] document in situ formation of SW dislocation dipoles. TEM measures also evidence [21] stable grain boundaries with alternating sequence of pentagons and heptagons that show the relevance of wave-like defects during graphene edge reconstruction.

_{6/6}(Figure 2a) of the chemical bond (arrowed) shared by the two hexagons creates the two 5|7 pairs (the SW defect 5/7/7/5). The second operator SW

_{6/7}turns the bond between the heptagon and the nearby shaded hexagon and inserts the 6|6 couple of shaded hexagons between the two original 5|7 pairs (this topological defect is also referenced in [4] as 5/7/6/6/7/5), leading to the overall structural effect of initiating the propagation of the SW wave along the dotted direction (Figure 2b). Iterated transformations SW

_{6/7}will successively drift the 5|7 pairs in the lattice (along the dotted directions in (Figure 2b), producing the topological SW wave (SWW).

_{p/r}rotations to just to the operators SW

_{6/6}and SW

_{6/7}. In spite of the apparent simplicity of our model, SW waves present an evident and marked topological anisotropy immediately signaled by the Wiener index [23] W(N) of the graphenic system under study (graphene fragments, CNT’s and GNR’s).

## 2. Generation and Propagation of Stone-Wales Rearrangements

_{p/r}simply rotates the internal edge between the p- and the r-connected nodes, making the study of the SW rearrangement very simple and suitable for automatic procedures.

_{6/6}rotation (Figure 2c) then changes four 6-connected nodes (white circles) into two 5-connected (shaded circles) and two 7-connected (black circles) vertices, matching the standard transformation in the direct lattice of four hexagons in two pentagons and two heptagons (Figure 2a). Moreover 5|7 pairs may also migrate in the graphene lattice, pushed by consecutive Stone-Wales transformations of SW

_{6/7}type that rotate, in the dual space, the vertical edge between the 6-, and the 7-connected vertices, driving the diagonal diffusion of a 5|7 pair in the graphene lattice. Figure 2d gives more details about the swapping mechanism between the 5|7 and the 6|6 couples. The repeated action of the SW

_{6/7}operator originates the topological SW wave in both lattice representations.

_{6/7}rearrangements, evidencing with the dashed arrows the increasing distance between the two 5|7 pairs of the original SW

_{6/6}dislocation. At each step, the pentagon (shaded circle) and the heptagon (black circle) interchange their locations with those of two hexagons (white circles) producing the diagonal SW wave, a large dislocation dipole that modifies the landscape of direct and dual lattices (Figures 4a and 4b). Being η the size of the dislocations (e.g., η equals the number of 6|6 pairs included between the two 5|7 pairs) both examples in Figure 4 have size η = 4, assuming size η = 0 for the basic SW

_{6/6}rotation of (Figures 1b). Equivalently, η equals the number of SW

_{6/7}rearrangements used to generate the dislocations in both spaces. A SW wave produces (Figure 4b) a characteristic hexagonal inter-grain spacing, isomeric to the pristine graphene layer that represents therefore a good theoretical model for the boundary between graphenic fragments.

_{p/r}to create novel classes of isomeric rearrangements, with rings made of various numbers of atoms, of fullerene (dimensionality D = 0), nanotubes (D = 1), graphenic structures (D = 2) or crystals (D = 3) as schwarzites or zeolites.

_{6/7}rotations.

- Diagonal SW wave, Figure 4: SW
_{6/7}rotates the vertical bond of the graphene dual lattice between the 6-connected node and the 7-connected node of the diffusing 5|7 pair, causing the diagonal drift of the pair and the creation of a new horizontal hexagon-hexagon bond (Figure 2d gives some more details); - Vertical SW wave, Figure 5: SW
_{6/7}rotates the diagonal bond of the graphene dual lattice between the 6-connected node and the 7-connected node of the diffusing 5|7 pair, with the overall effect to vertically shift the pair, generating a new anti-diagonal hexagon-hexagon bond.

_{6/6}rearrangement. In the following we mainly study this latter case, focusing on the mechanisms leading to the creation of diagonal or vertical extended dislocation dipoles in the graphene lattice.

_{n}fullerene starting for a limited number of inequivalent cages.

## 3. Theoretical Basis of the Topological Model

_{60}fullerene isomers, just the physically stable isomer with icosahedral symmetry C

_{60}-I

_{h}and isolated pentagons corresponds to the isomer with the minimum W value W = 8340 and the highest topological compactness. This concept is naturally extended to the isomers of any other carbon chemical systems. Our simulations aim therefore for the most-compact structures seen as very good candidates for chemically stable systems. [29,31] present recent successful applications of this method to graphenic layers and C

_{66}fullerene. Computationally, our assumption implies the topological minimum principle on W: chemically stable structures have to be searched among the configurations minimizing the W index.

_{66}fullerene [31] with a good match with the experimental results.

_{ij}between all pairs of vertices V

_{i}, V

_{j}in the lattice:

^{3}, being that a particular case of the polynomial-like general formula W(N) ≈ N

^{s}(with s =2 + 1/D) recently conjectured for large D-dimensional lattices [29]. In case of D = 2 structures, the general closed form for the Wiener index is:

## 4. Results and Discussions

_{10}dual graphene lattice consisting of 10 ×10 unit cells with N = 200 starred vertices and periodic boundary conditions. Equation (3) attributes to that ideal closed lattice G

_{10}the reference value of the topological potential W

_{G10}= 116,500.

_{6/6}flip produces the 5|7 double pair (Figures 2a and 2c) and decreases the lattice potential to W = 116,015, easily derivable from the direct computation of the graph chemical distances according to the definition (1). In our approximated model this negative 0.42% variation of the Wiener index represents the topological gain induced by the creation of a diagonally oriented SW defect.

_{6/7}rotation (Figures 2b and 2d) translates one of the 5|7 pair with a further decrease of the topological potential being W = 115 870 for the step η = 1 on G

_{10}. This behavior is confirmed at each propagation steps of the diagonal SW wave, augmenting the topological stability of the system. The reduction of the topological potential W follows an almost linear trend, see the top curve of (Figure 6a) where the number of propagation steps η is reported, the η = 0 case corresponding to the creation of the diagonal dislocation SW dipole with topological potential W = 116,015. The result evidences the tendency of the 10 × 10 graphene fragment to allow the unlimited topological diffusion the 5|7 pair along the diagonal direction with the creation of extended diagonal dislocation dipoles (Figures 4a and 4b). This characteristic of the diagonal SW wave has a pure topological root strongly correlated to the connectivity properties of the pentagon-heptagon pairs embedded in the hexagonal mesh and to the edge effect induced by the fragment boundary.

_{6/6}vertical rotation, the two new 5|7 pairs give W = 116,425 for η = 0 with a little topological gain of just −0.06% compared to W

_{G10}= 116,500 of the pristine lattice, smaller than the one (−0.42%) detected in the diagonal case. The successive SW

_{6/7}flip starts the vertical propagation (η = 1) of one of the 5|7 defect, slight increasing the topological potential W = 116,426. This growth of the topological potential opposes to the vertical diffusion of the wave, being this barrier effect confirmed at the successive steps η = 2 and η = 4 by the increasing values W = 116,438 and W = 116,455. According to our simulations therefore the topological potential W obstacles (Figure 6a bottom) the diffusion of the vertical SW wave in the closed 10 × 10 graphene fragment G

_{10}.

_{25}with N = 1250 starred nodes and starting value W

_{G25}= 11,390,625. After the first SW diagonal rotation (η = 0) the potential passes to W

_{η}

_{=0}= 11,373,153 with a gain in the topological stability of about −0.15%. One may also observe the vertical propagation on the G

_{25}lattice suffuses that of G

_{10}one in accordance with the idea that for infinite extended system it will resemble the potential well, within which the diagonal movement takes place. The N

_{5/2}leading terms in Equation (3) give for the ideal lattices with N = 1250 and N = 200 an approximated ratio of 97.66 that matches quite well the corresponding fraction W

_{G25}/W

_{G10}≈ 97.77 showing the fast convergence of the Wiener index polynomial (3).

_{10}an G

_{25}layers with one SW defect (η = 0) suggesting that an infinite set of exact polynomial functions W(N, η) may be found as generalization of the Formula (3) to describe—still with the N

_{5/2}dependence—the topological potential W in presence of diagonal (or vertical) dislocation dipoles with variable size η; this topological property will be the subject of future investigations.

_{25}lattice confirm that the diagonal propagation on large distances of the 5|7 pair is still favored (Figure 6b top) whereas just a limited penetration (η = 5) of the vertical SW dislocation dipole is allowed (Figure 6b bottom). This behavior differs from the sharp potential barrier encountered by the vertical SW wave in the smaller G

_{10}layer (Figure 6a bottom) and one may consider the limited vertical propagation in G

_{25}as the tendency of the system to recover, for large N, the equivalency between the two plane directions that, for the infinite graphenic sheet, are totally indistinguishable.

- The tendency of the 5|7 defects [19,22] to cover large graphene regions find here a specific topological mechanism, the SW wave. It produces (via the initial SW
_{6/6}flip) two 5|7 pairs and then separates them via consecutive SW_{6/7}rotations, creating an extended dislocation dipole; the reversed isomeric operation may also take place to annihilate distant 5|7 pairs; - In the graphene layer SW
_{6/7}flips are also able to transport isolated 5|7 dislocation monopoles, by exchanging the heptagon and pentagon positions with those of two nearby hexagons; this drifting mechanism may also annihilate or modify the 5|7 pair in colliding with other structural defects (grain boundaries, other 5|7 pairs, generic q|r pairs, etc.). This result integrates previous studies [1] providing the invoked local annealing mechanism.

_{25}lattice or the 6344 atoms square graphite sheets previously modeled in literature using molecular mechanics tools [4]. In ref. [28] the characteristic spatial patterns of electric current flow are studied in metallic arm-chair CNT’s depending on the orientation of the SW defect. The presence of rotating loop currents at nanometer scale is originated by quantum interference of conducting and quasi-bound states of electrons in the region of the dislocation dipole, and generates typical patterns of induced magnetic dipoles suitable for experimental detection. The distribution of the loop currents effectively distinguishes the symmetry of the SW defects suggesting that this anisotropic magnetic effect may occur in a general nanostructure, finding potentially application in novel electronic and magnetic nanodevices. Electronic and chemical properties of 5|7 or 5/7/7/5 topological defects are different from the ones exhibited by structural defects (e.g., the presence of single non-hexagonal rings surrounded by hexagonal rings) and, according to review [3], “their reactivity and detection needs to be investigated theoretically and experimentally”. Article [21] gives an interesting evidence of a possible topological SWW mechanism, showing the linear defect that appears during the TEM edge reconstructions of a graphene sheet under the effect of a 80 kV transmission electron microscopy. The stable edge configuration, made of an alternating sequence of pentagons and heptagons, swaps with the pristine zigzag edge, the energy input from the beam proving the required activation energy.

_{n}nanoflakes in which 5|7 pairs are stable defects according to energy-minimization technique [34].

## 5. Conclusions

_{q}

_{|}

_{r}rotations to produce an endless, at the moment largely unknown, sequence of new isomeric configurations in chemical structures with various dimensionality like fullerenes, nanotubes, graphenic layers, schwarzites, zeolites.

## Acknowledgements

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**Figure 1.**(

**a**) Local transformation SW

_{q/r}changes a group of four proximal faces with p, q, r, s atoms in four new rings with p−1, q+1, r−1, s+1 atoms; (

**b**) On the graphene layer (p=q=r=s=6) SW

_{6/6}reversibly flips four hexagons in a 5|7 double pair; (

**c**) SW

_{5/6}reversible flip on the fullerene surface.

**Figure 2.**(

**a**) SW

_{6/6}originates two 5|7 pairs (in gray); (

**b**) SW

_{6/7}splits the pairs by swapping one of them with two nearby hexagons (shaded). Dotted SW

_{6/7}pushes the SW wave in the dashed direction; (

**c-d**) Mechanisms (a,b) in the graphene dual plane. Hexagons, pentagons, heptagons are represented by white, shaded, black circles respectively.

**Figure 3.**Dual representation of the graphene lattice obtained by replacing each hexagonal face by the central 6-connected graph node. Graphene plane is then equivalently tiled by hexagons (direct space) or by starred nodes (dual space). The x-periodic (y-periodic) direct graphene nanoribbon has the armchair (zig-zag) orientation. The framed unit cell has been used to build this 4 × 7 graphenic fragment.

**Figure 4.**(

**a**) Diagonal SW wave (dislocation dipole) in the dual graphene layer after the generation and four propagation steps (size η = 4); at each step (dashed arrows) SW

_{6/7}swaps the pair made by one pentagon (dashed circle) and one heptagon (black circle) with two connected hexagons (dotted circles); dotted arrow indicates the next available translation of the 5|7 pair; (

**b**) The topological modification (a) originates, in the direct lattice, a hexagonal inter-grain spacing (dashed rings); (

**c**) After a few more SW rotations, an isolated pentagon (arrowed), forming a small nanocone, is generated.

**Figure 5.**SW vertical wave in the dual graphene layer after four propagation steps (dashed arrows); SW

_{6/7}swaps the pentagon (dashed circle) heptagon (black circle) pair with two hexagons (dotted circles); dotted arrow indicates the next possible translation of the 5|7 pair, induced by a SW

_{6/7}rotation of the hexagon-heptagon diagonal dashed bond. The SW wave generates anti-diagonal hexagons-hexagons bonds with respect to the unrotated one.

**Figure 6.**(

**a**) Wiener index W for diagonal and vertical SW waves for the N = 200 dual closed graphene graph G

_{10}as a functions of the wave propagation steps η. Diagonal dislocation dipoles (top) freely flow in the lattice, whereas the vertical ones (bottom) are stopped; (

**b**) On the N = 1250 lattice G

_{25}, vertical SW waves (bottom) present a limited penetration (η = 6), being the diagonal penetration of the defects still favored (top). The ratio W

_{G}

_{25}/W

_{G}

_{10}≈ 97.77 between the W values for the two ideal lattices W

_{G}

_{25}= 11390625 and W

_{G}

_{10}= 116500 follows the ratio of the N

_{5/2}leading terms in Equation (3).

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Ori, O.; Cataldo, F.; Putz, M.V.
Topological Anisotropy of Stone-Wales Waves in Graphenic Fragments. *Int. J. Mol. Sci.* **2011**, *12*, 7934-7949.
https://doi.org/10.3390/ijms12117934

**AMA Style**

Ori O, Cataldo F, Putz MV.
Topological Anisotropy of Stone-Wales Waves in Graphenic Fragments. *International Journal of Molecular Sciences*. 2011; 12(11):7934-7949.
https://doi.org/10.3390/ijms12117934

**Chicago/Turabian Style**

Ori, Ottorino, Franco Cataldo, and Mihai V. Putz.
2011. "Topological Anisotropy of Stone-Wales Waves in Graphenic Fragments" *International Journal of Molecular Sciences* 12, no. 11: 7934-7949.
https://doi.org/10.3390/ijms12117934