First-Principles Study of Graphene and Carbon Nanotubes

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The Journal of Physical Chemistry

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First-Principles Study of Graphene and Carbon 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3

Nanotubes Functionalized with Benzyne Mahmoud Hammouri, Sanjiv K. Jha, and Igor Vasiliev∗ Department of Physics, New Mexico State University, Las Cruces, New Mexico 88003 E-mail: [email protected]

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To whom correspondence should be addressed

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1 ACS Paragon Plus Environment


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Abstract

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We applied ab initio computational methods based on density functional theory 6 7

to study the properties of graphene and single-walled carbon nanotubes functional8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

ized with benzyne. The calculations were carried out using the SIESTA electronic structure code combined with the generalized gradient approximation for the exchange correlation functional. Our study showed that the reaction of cycloaddition of benzyne to pristine graphene was exothermic with the possibility of formation of both [2+2] and [4+2] reaction products. The binding energies of benzyne molecules attached to semiconducting zigzag and metallic armchair nanotubes were found to be inversely proportional to the nanotube diameter. The linear fits of the binding energies between benzyne and carbon nanotubes extrapolated to the zero curvature limit were in good agreement with the binding energies of benzyne attached to graphene. Our calculations demonstrated that the cycloaddition of benzyne could open up a nonzero gap between the valence and conduction bands of graphene and metallic carbon nanotubes. The value of the band gap was significantly affected by the geometry of benzyne attachment and by the choice of the supercell.

Keywords graphene, carbon nanotubes, benzyne, functionalization, density functional theory

Introduction 45 46 47

Carbon nanomaterials have attracted considerable attention in recent years. The remarkable 48 49 50 5 1 5 2 5 3

properties of graphene and carbon nanotubes (CNTs) make them promising candidates for

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functionalization expands the range of possible applications of carbon nanomaterials. Surface

a variety of applications in electronics, chemistry, and materials engineering. 1,2 Chemical

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functionalization increases the solubility of graphene and CNTs in organic solvents, opening 56 57

a way for the synthesis of polymer-carbon nanocomposites. 3–6 Covalent functionalization 58 59 60


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alters the electronic and transport properties of carbon nanostructures, providing a basis for the development of molecular electronics. 7–10

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However, in the absence of surface defects, carbon nanomaterials are chemically inert and 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

do not easily interact with molecules and functional groups. The low reactivity of graphene and CNTs emphasizes the need for finding chemically active agents that can be covalently bound to the surfaces of carbon nanostructures. Up to date, only a limited number of chemical reactions leading to covalent functionalization of pristine non-oxidized graphene and CNTs has been observed in experiments. 11–21 Recent experimental studies have shown the possibility of cycloaddition of arynes, such as benzyne, to the surface of pristine graphene 17,21 and to the sidewalls of CNTs. 16,19 The successful functionalization of graphene and CNTs with benzyne has been confirmed by X-ray photoelectron spectroscopy (XPS), energy dispersive X-ray (EDX) spectroscopy, fourier transform infrared spectroscopy (FTIR), Raman spectroscopy, high-resolution transmission electron microscopy (HRTEM), and thermogravimetric analysis (TGA). 16,17,21 These studies demonstrated that aryne cycloaddition provides an efficient method of chemical modification of pristine carbon nanomaterials under mild reaction conditions. Despite significant interest in covalent functionalization of carbon nanostructures, the mechanism of cycloaddition of arynes to pristine carbon nanomaterials has not been systematically investigated. Previous theoretical studies of cycloaddition of benzyne to graphene and CNTs have focused on several selected structures. 19,22,23 Denis and Iribarne investigated the interaction of benzyne with graphene, but only the [2+2] cycloaddition reaction was

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considered in their study. 22 Criado et al. computed the lowest energy structures for three 47 48

zigzag [(10,0), (14,0), (18,0)] and two armchair [(7,7), (10,10)] CNTs functionalized with 49 50 5 1 5 2 5 3

benzyne. 19 Yang et al. studied the reaction mechanism of benzyne cycloaddition to (n,n)

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(n,0) zigzag CNTs with n = 7−12. 23 The results of these studies indicated that the topology 5 56

armchair CNTs with n = 5 − 12 and calculated the binding energies for benzyne attached to

of cycloaddition of benzyne to CNTs was sensitive to the size and type of the CNT, with the 57 58 59 60



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possibility of formation of both [2+2] and [4+2] reaction products. 19,23 The studies of Denis and Iribarne, 22 Criado et al., 19 and Yang et al. 23 were performed using different computa-

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tional methods, which made it difficult to compare the calculated properties of graphene 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3

and CNTs functionalized with benzyne. Furthermore, in the calculations of Criado et al. and Yang et al., CNTs were modeled using hydrogen-passivated CNT fragments of finite length. This computational model did not take into account the influence of metallic and semiconducting electronic properties on the surface reactivity of armchair and zigzag CNTs. In our paper, we presented a comprehensive first principles density functional study of cycloaddition of benzyne to pristine graphene and single-walled CNTs. The computed structures, binding energies, and band structures of CNTs and graphene functionalized with benzyne were compared with the results of previous theoretical and experimental studies. Based on our calculations, we examined the mechanisms of cycloaddition of benzyne to graphene, zigzag CNTs, and armchair CNTs and analyzed the dependence of the binding energies between CNTs and benzyne on the type and size of the CNT. We also investigated the influence of benzyne cycloaddition on the electronic properties of CNTs and graphene.

Computational Methods Our computational approach was based on density functional theory (DFT) combined with the pseudopotential approximation. 24,25 Calculations were carried out using the SIESTA (Spanish Initiative for Electronic Simulations with Thousands of Atoms) electronic struc-

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ture code. 26,27 Within the SIESTA code, electronic wave functions were expanded in a basis 46 47

of Gaussian-type atomic orbitals. We employed the Kleinman-Bylander form 28 of norm48 49 50 5 1 5 2 5 3

conserving Troullier-Martins pseudopotentials. 29 The exchange-correlation energy functional

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Burke, and Ernzerhof. 30 Computational methods based on DFT combined with the GGA 5 56

was evaluated using the generalized gradient approximation (GGA) parameterized by Perdew,

functional have been successfully used in the past to study the properties of functionalized 57 58 59 60


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graphene and CNTs. 22,31–36 To improve the accuracy of the computed binding energies and equilibrium interatomic distances, all calculations were conducted using the double-ζ plus

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polarization (DZP) basis set. 37 The Hartree and exchange-correlation potentials were com8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

puted by projecting the electronic density and wave functions onto a real-space grid. The energy cut-off, which corresponded to the grid spacing, was set to 400 Ry for graphene and to 270−370 Ry for CNTs. The mechanism of surface functionalization of carbon nanomaterials with benzyne is shown in Fig. 1. The numbers 1 through 6 indicate the different positions of benzyne molecules attached to graphene, zigzag CNTs, and armchair CNTs. The structures of graphene sheets and CNTs functionalized with benzyne were obtained using a periodic supercell method within the SIESTA code. Graphene sheets were modeled by 6×6 and 8×8 hexagonal supercells containing 72 and 128 carbon atoms, respectively. The Brillouin zone was sampled with a Monkhorst-Pack 12×12×1 k-point mesh for the 6×6 graphene supercell and a 8×8×1 k-point mesh for the 8×8 graphene supercell. Zigzag and armchair CNTs were modeled by periodic supercells containing 3 and 4 unit cells, respectively. The Brillouin zone was sampled with a 4×1×1 k-point mesh for semiconducting zigzag CNTs and a 20×1×1 k-point mesh for metallic armchair CNTs. To eliminate the effects of artificial periodicity, the dimensions of the supercells in the directions orthogonal to the graphene plane and to the axis of the CNT were selected to be greater than the cut-off radii of the SIESTA basis orbitals. Structural optimization of pristine and functionalized carbon nanostructures was per-

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formed via the conjugate gradient method. The residual interatomic forces were required 47 48

to be smaller than 0.03 eV/˚ A. For pristine graphene, our SIESTA calculations predicted an 49 50 5 1 5 2 5 3

equilibrium C−C bond length of 1.44 ˚ A, which was in good agreement with the experimental value of 1.42 ˚ A for graphite and graphene.

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Results and Discussion

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Cycloaddition of Benzyne to Graphene 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

First, we examined the cycloaddition reaction of benzyne to pristine graphene. The optimized structures of graphene sheets functionalized with benzyne in the 1-2 [2+2], 1-3 [3+2], and 1-4 [4+2] attachment configurations are shown in Fig. 2. We found that the interaction with benzyne induced significant structural changes in graphene. The attached benzyne molecule was almost orthogonal to the surface of graphene. After the attachment of benzyne to graphene, the C−C bonds near the functionalization sites increased in length from 1.44 ˚ A to 1.51−1.52 ˚ A, whereas the the bond angles decreased from 120◦ to 112◦ −116◦ . The reduction of the angles between the C−C bonds indicated a significant buckling of the graphene layer near the functionalization sites. The buckling could be explained by a partial change of hybridization of the C atoms at the functionalization sites from sp2 to sp3 and the creation of new covalent bonds between benzyne and graphene. The structural changes observed in graphene functionalized with benzyne were similar to those previously reported in graphene functionalized with hydrogen, hydroxyl, carboxyl, epoxy, nitrene, and methyl chemical groups. 32,34,35,41–45 The calculated binding energies, Eb , and equilibrium C−C bond lengths, deq , for benzyne molecules attached to pristine graphene are shown in Table 1. The equilibrium C−C bond lengths between benzyne and graphene were defined as the distances between the C atoms of the benzyne molecule and the C atoms of graphene at the functionalization sites. The

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binding energies between benzyne and graphene were evaluated as Eb = ECn +C6 H4 − ECn − 47 48

EC6 H4 , where EC6 H4 , ECn , and ECn +C6 H4 were the total energies of an isolated benzyne 49 50 5 1 5 2 5 3

molecule, pristine graphene, and functionalized graphene, respectively. Negative values of Eb corresponded to thermodynamically stable structures. To eliminate the localized basis

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set superposition error (BSSE) in SIESTA calculations, the binding energies were computed 5 56

using the counterpoise correction method proposed by Boys and Bernardi. 38 Within this 57 58 59 60


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method, the BSSE was corrected by including additional wave functions without any atomic potential into the basis set.

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The results of our calculations indicated a good convergence of the computed structures 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

and binding energies with respect to the size of the graphene supercell. The interaction energies between benzyne and graphene computed using the 6×6 and 8×8 hexagonal supercells agreed to within 0.01−0.04 eV and the equilibrium bond lengths were almost identical. Our study demonstrated that the reactions of 1-2 [2+2] and 1-4 [4+2] cycloaddition of benzyne to graphene were exothermic with the binding energies of −0.73 eV and −0.58 eV, respectively. The 1-2 [2+2] attachment was found to be ∼0.15 eV lower in energy than the 1-4 [4+2] attachment. A relatively small difference between the energies of these structures suggested that both the [2+2] and [4+2] cycloaddition reactions between benzyne and graphene may be observed in experimental studies. 21 In contrast, the reaction of 1-3 [3+2] cycloaddition of benzyne to pristine graphene was found to be endothermic with the binding energy of +0.44 eV. The results of our calculations for graphene functionalized with benzyne were consistent with previous observations that arynes can react with polycyclic aromatic compounds by [2+2] or [4+2] cycloadditions, whereas the [3+2] cycloaddition is expected to be energetically unfavorable. 39,40 Our calculated binding energy for the [2+2] cycloaddition of benzyne to pristine graphene was in good agreement with a previous theoretical study performed by Denis and Iribarne, which predicted the binding energies in the range from −12.2 kcal mol−1 (−0.53 eV) to −16.3 kcal mol−1 (−0.71 eV) using the 4×4 and 5×5 hexagonal graphene supercells combined with several different basis sets. 22

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The calculated electronic band structures of pristine graphene and graphene functional47 48

ized with benzyne in the 1-2 [2+2] and 1-4 [4+2] attachment positions are shown in Fig. 3. 49 50 5 1 5 2 5 3 54

The energy bands were computed using the 6×6 and 8×8 hexagonal graphene supercells. The Γ, K, and M points in the energy band plots correspond to the center, the corner, and the edge center of the first Brillouin zone, respectively. Pristine graphene is a zero-gap

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material. The valence and conduction bands of pristine graphene have a conical shape and 57 58 59 60



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touch each other at the Dirac points located at the K and K’ corners of the first Brillouin zone. 46 In the 6×6 and 8×8 hexagonal graphene supercells, the Dirac points are located at

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the Γ and K points, respectively. Our calculations demonstrated that the cycloaddition of 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

benzyne opened up a nonzero gap between the valence and conduction bands of graphene. We also found that cycloaddition of benzyne to graphene produced several new energy bands located above and below the Fermi level. These bands were associated with the electronic states resulting from the interaction between benzyne molecules and graphene. Table 2 presents the calculated direct band gaps at the Γ and K points and the minimum band gaps for graphene functionalized with benzyne in the 1-2 [2+2] and 1-4 [4+2] attachment positions. The band gaps were computed for n × n hexagonal graphene supercells with n = 3 − 9. In supercells in which n was not a multiple of 3, the Dirac point in pristine graphene was located at the K point of the first Brillouin zone. However, in supercells in which n was a multiple of 3, the Dirac point in pristine graphene appeared at the Γ point due to the projection of the K point to the center of the first Brillouin zone. Our calculations showed that the cycloaddition of benzyne to graphene opened up a nonzero gap at the Γ point in graphene supercells with n = 3, 6, and 9. The band gap increased with decreasing the size of the graphene supercell and increasing the effective concentration of benzyne molecules attached to the surface of graphene. A similar relationship between the size of the band gap and the effective concentration of adsorbates has previously been reported for graphene functionalized with other chemical groups. 31,33,43,47,48 We also observed the opening of a nonzero gap at the K point in hexagonal supercells in which n was not a multiple of 3.

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However, in these supercells the valence and conduction bands of graphene functionalized 47 48

with benzyne touched each other between the Γ and K, or K and M points. Consequently, 49 50 5 1 5 2 5 3 54

the minimum value of the band gap was equal to zero in supercells with n = 4, 5, 7, and 8. 22 Our study indicated that the choice of the supercell significantly affected the calculated band gap values in graphene functionalized with benzyne. The results of our calculations

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were consistent with a previous theoretical study of graphene functionalized with hydrogen 57 58 59 60


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atoms, which demonstrated a strong dependence of the band gap opening at the Dirac point of graphene on the size of the supercell and the periodicity of the adsorbed hydrogen atoms. 49

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To test the influence of inhomogeneous distribution of the attached benzyne molecules 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

on the electronic properties of graphene, we computed the band structures of graphene functionalized with two benzyne molecules per supercell. The benzyne molecules were attached to the opposite sides of the graphene sheet. The energy bands were computed using the 5×5 hexagonal graphene supercell. The calculated electronic band structures of graphene functionalized with two benzyne molecules in the 1-2 [2+2] and 1-4 [4+2] attachment positions are shown in Fig. 4. The computed band gaps are presented in the last line of Table 2. Our calculations indicated that the addition of the second benzyne molecule per supercell did not fundamentally change the band structure of functionalized graphene. Similarly to the case of the 5×5 graphene supercell functionalized with one benzyne molecule, we observed the opening of a nonzero gap at the K point, whereas the minimum value of the band gap was equal to zero due to the fact that the valence and conduction bands touched each other between the Γ and K, or K and M points. The computed densities of states (DOS) of graphene functionalized with benzyne in the 1-2 [2+2] and 1-4 [4+2] attachment positions are shown in Fig. 5. The DOS plots were obtained using a 50×50×1 k-point sampling of the 6×6 hexagonal graphene supercell. For comparison, the DOS of pristine graphene was also included in the plots. The zero energy in Fig. 5 was set at the Fermi level. We found that near the Fermi level the DOS curves of pristine graphene and graphene functionalized with benzyne were similar to each other.

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The position of the Dirac point in graphene functionalized with benzyne was not noticeably 47 48

changed compared to that in pristine graphene. At the same time, the positions of the van 49 50 5 1 5 2 5 3 54

Hove peaks 50 in the DOS of graphene functionalized with benzyne were shifted compared to that in pristine graphene due to structural distortions of the carbon lattice. The DOS of functionalized graphene also exhibited several additional peaks located above and below

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the Fermi level. These peaks were associated with the new energy bands produced by the 57 58 59 60



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cycloaddition of benzyne to graphene.

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Cycloaddition of Benzyne to Carbon Nanotubes 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

Next, we examined the cycloaddition reaction of benzyne to carbon nanotubes. The equilibrium structures of the (8,0) zigzag and (5,5) armchair CNTs functionalized with benzyne in different attachment positions are shown in Figs. 6 and 7, respectively. The 3-6 [4+2] attachment configuration for the (5,5) CNT was found to be unstable and spontaneously converged into the 1-2 [2+2] attachment configuration. Overall, the changes observed in the geometries of CNTs functionalized with benzyne were consistent with the results of our calculations for graphene. After the attachment of benzyne to CNTs, the C−C bonds near the functionalization sites increased in length by 4% to 5%, whereas the the bond angles decreased by 3◦ to 6◦ . The increase of the bond lengths and the reduction of the angles between the C−C bonds could be explained by the change of hybridization of the C atoms at the functionalization sites from sp2 to sp3 due to the formation of new covalent bonds between benzyne and CNTs. Table 3 presents the calculated binding energies, Eb , and equilibrium C−C bond lengths, deq , for benzyne molecules attached to the (8,0) zigzag and (5,5) armchair CNTs. The equilibrium C−C bond lengths between benzyne and CNTs were defined as the distances between the C atoms of the benzyne molecule and the C atoms of CNTs at the functionalization sites. The binding energies between benzyne and CNTs were evaluated as Eb = ECN T +C6 H4 − ECN T − EC6 H4 , where EC6 H4 , ECN T , and ECN T +C6 H4 were the total en-

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ergies of an isolated benzyne molecule, pristine CNT, and functionalized CNT, respectively. 48 49 50 5 1 5 2 5 3 54 5

Our calculations demonstrated that the interaction energies between benzyne and CNTs strongly depended on the relative orientation of the benzyne molecule with respect to the longitudinal axis of the CNT. The cycloadditions of benzyne to the (8,0) zigzag CNT favored the attachment configurations in which the plane of the benzyne molecule was parallel

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to the axis of the CNT, whereas the [2+2] cycloaddition of benzyne to the (5,5) armchair 58 59 60


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CNT favored the attachment configuration in which the plane of the benzyne molecule was orthogonal to the axis of the CNT. According to our study, the 1-2 [2+2] cycloaddition of

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benzyne to the (8,0) and (5,5) CNTs produced the most stable reaction products. For the 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

(5,5) armchair CNT, the 1-2 [2+2] attachment configuration was ∼0.9 eV lower in energy than the 1-4 [4+2] attachment configuration. For the (8,0) zigzag CNT, the energies of the 1-2 [2+2] and 3-6 [4+2] benzyne attachments were almost identical. The absolute values of the binding energies between benzyne and CNTs were found to be significantly larger than those between benzyne and graphene. It is interesting to note that our study predicted the possibility of exothermic [3+2] cycloaddition reactions of benzyne with the (8,0) and (5,5) CNTs. The results of our study indicated that the sidewalls of CNTs were significantly more reactive toward the cycloaddition of arynes than the surface of pristine graphene. Figure 8 shows the calculated electronic band structures of the (8,0) zigzag and (5,5) armchair CNTs functionalized with benzyne in the most stable [2+2] and [4+2] attachment positions. The energy bands were computed using the periodic supercells containing 3 unit cells for the (8,0) CNT and 4 unit cells for the (5,5) CNT, respectively. The zero energy in Fig. 8 corresponded to the Fermi level. The valence and conduction bands of the semiconducting (8,0) CNT are separated by a relatively large gap, whereas the metallic (5,5) CNT has a zero gap between the valence and conduction bands at the Fermi level. Our calculations demonstrated that the attachment of benzyne reduced the size of the band gap of the semiconducting (8,0) CNT. In contrast, we found that the cycloaddition of benzyne in the 1-4 [4+2] attachment position opened up a gap between the valence and conduction bands of

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the metallic (5,5) CNT. At the same time, the effect of the cycloaddition of benzyne in the 47 48

1-2 [2+2] attachment position on the band structure of the (5,5) CNT was relatively small. 49 50 5 1 5 2 5 3 54

Similarly to the case of graphene, the functionalization of CNTs with benzyne produced several new energy bands associated with the electronic states resulting from the interaction between benzyne molecules and the CNTs. The appearance of impurity bands has previously

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been observed in CNTs functionalized with other chemical groups. 36 57 58 59 60



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The calculated band gaps of the (8,0) zigzag and (5,5) armchair CNTs functionalized with benzyne in different attachment positions are plotted in Fig. 9 as a function of the number

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of CNT unit cells. Our calculations showed that the influence of chemical functionalization 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

on the electronic properties of the (8,0) and (5,5) CNTs became more pronounced with decreasing the size of the supercell and increasing the effective concentration of the attached benzyne molecules. The computed band gaps of the functionalized metallic (5,5) CNT strongly depended on the position and orientation of the attached benzyne molecule. For the 1-6 [2+2] and 1-4 [4+2] attachments of benzyne to the (5,5) CNT the band gaps were in the range of ∼0.1−0.4 eV, whereas for the 1-2 [2+2] attachment configuration the band gap was close to zero. The results of our calculations were consistent with a previous theoretical study, which predicted a strong dependence of the band gap opening in metallic CNTs functionalized with various chemical groups on the positions of the chemisorption sites on the surface of the CNT. 51 The calculated binding energies, Eb , for semiconducting zigzag CNTs of different diameters, D, functionalized with benzyne in the most stable attachment positions are presented in Table 4. For all studied zigzag CNTs, the 1-6 [2+2] and 1-4 [4+2] attachments of benzyne were found to be significantly higher in energy than the 1-2 [2+2] and 3-6 [4+2] attachments. At the same time, our calculations revealed that the energies of the 1-2 [2+2] and 3-6 [4+2] cycloadditions of benzyne to zigzag CNTs were close to each other and differed by less than 0.15 eV. A relatively small difference between the energies of these structures suggested that both the [2+2] and [4+2] cycloaddition reactions between benzyne and zigzag CNTs may be

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observed in experimental studies. We found that for the (5,0), (7,0), and (16,0) CNTs, the 347 48

6 [4+2] attachments of benzyne were slightly lower in energy than the 1-2 [2+2] attachments. 49 50 5 1 5 2 5 3 54

In contrast, for the (8,0), (11,0), and (14,0) CNTs, the 1-2 [2+2] benzyne attachments were slightly lower in energy than the 3-6 [4+2] attachments. Our computed binding energies for the 1-2 [2+2] and 3-6 [4+2] cycloadditions of benzyne to (7,0), (8,0), and (11,0) zigzag

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CNTs were in reasonably good agreement with a previous theoretical study carried out by 57 58 59 60


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Yang et al. 23 Overall, our binding energies for the attachments of benzyne to zigzag CNTs were approximately 0.3−0.5 eV lower than the corresponding values of Yang et al.

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The calculated binding energies, Eb , for metallic armchair CNTs of different diameters, 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

D, functionalized with benzyne in the most stable attachment positions are shown in Table 5. Our calculations demonstrated that for all studied armchair CNTs, the 1-2 [2+2] and 1-6 [2+2] cycloadditions of benzyne were significantly lower in energy than the 1-4 [4+2] cycloadditions. Our study predicted that for the (3,3), (4,4), and (5,5) CNTs, the 1-2 [2+2] attachments of benzyne were lower in energy than the 1-6 [2+2] attachments. In contrast, we found that for the (8,8) CNT, the 1-2 [2+2] benzyne attachment was slightly higher in energy than the 1-6 [2+2] attachment. Our computed binding energies for the 1-2 [2+2], 1-6 [2+2], and 1-4 [4+2] cycloadditions of benzyne to (5,5) and (8,8) armchair CNTs for the most part agreed with the results of the previous theoretical study performed by Yang et al. 23 Overall, our computed binding energies for the attachments of benzyne to armchair CNTs were approximately 0.2−0.7 eV lower than the corresponding values obtained by Yang et al. The discrepancy between our binding energies and those of Yang et al. could be attributed to the difference in the computational models: in our calculations the structures of CNTs were modeled using periodic supercells, whereas in the study of Yang et al. CNTs structures were modeled using hydrogen-passivated CNT fragments. The variations of the calculated binding energies, Eb , for semiconducting zigzag CNTs functionalized with benzyne in the most stable 1-2 [2+2] and 3-6 [4+2] attachment positions are plotted in Fig. 10 as functions of the CNT inverse diameter. The dashed lines show the

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best linear fits of Eb vs 1/D. Our calculations demonstrated that the surface reactivity of 47 48

zigzag CNTs toward the cycloaddition of benzyne increased with decreasing the diameter of 49 50 5 1 5 2 5 3 54

the CNT and increasing the curvature of the CNT sidewall. The binding energies of benzyne molecules attached to semiconducting zigzag CNTs were found to be inversely proportional to the CNT diameter. A similar linear dependence of Eb on 1/D has previously been observed

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for CNTs functionalized with hydroxyl, amine, and carboxyl chemical groups. 52 The results 57 58 59 60



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of our study indicated that the linear fits of Eb vs 1/D extrapolated to the limit of D → ∞ were in very good agreement with the computed binding energies of benzyne attached to

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pristine graphene in the 1-2 [2+2] and 1-4 [4+2] positions. 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3

Figure 11 shows the variations of the calculated binding energies, Eb , vs the CNT inverse diameter for metallic armchair CNTs functionalized with benzyne in the 1-2 [2+2], 1-6 [2+2], and 1-4 [4+2] attachment positions. We found that, similarly to the case of zigzag CNTs, the surface reactivity of armchair CNTs toward benzyne increased with decreasing the diameter of the CNT and increasing the curvature of the CNT sidewall. The absolute values of the binding energies of benzyne molecules attached to metallic armchair CNTs in the 1-6 [2+2] and 1-4 [4+2] positions increased in inverse proportion to the CNT diameter. At the same time, we observed deviations from the linear relationship between Eb and 1/D for the 1-2 [2+2] attachments of benzyne to armchair CNTs. The observed differences in the dependence of Eb on 1/D for the 1-2 [2+2], 1-6 [2+2], and 1-4 [4+2] cycloadditions of benzyne to armchair CNTs could be attributed to a geometrical factor. For armchair CNTs functionalized with benzyne in the 1-2 [2+2] attachment position, the plane of the benzyne molecule was orthogonal to the axis of the CNT. We found that the angles between the C−C bonds at the functionalization sites were more strongly affected by the curvature of the CNT sidewall when the benzyne molecule was orthogonal to the CNT axis. This observation could explain the nonlinear dependence of Eb on 1/D for the 1-2 [2+2] cycloaddition of benzyne to armchair CNTs.

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Conclusion 47 48 49 50 5 1 5 2 5 3

We applied first-principles density functional computational methods to study the cycload-

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using the SIESTA electronic structure code. The exchange-correlation energy functional 5 56

dition of benzyne to graphene and single-walled CNTs. Our calculations were performed

was evaluated using the generalized gradient approximation. Our study demonstrated that 57 58 59 60


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the reactions of [2+2] and [4+2] cycloaddition of benzyne to pristine graphene were exothermic. The sidewalls of CNTs were found to be significantly more reactive toward benzyne

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than the surface of graphene. The binding energies of benzyne molecules attached to semi8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4

conducting zigzag and metallic armchair CNTs were, in most cases, inversely proportional to the CNT diameter. Our calculations showed that for the semiconducting zigzag CNTs the [2+2] and [4+2] cycloadditions of benzyne were close to each other in energy, whereas for the metallic armchair CNTs the [2+2] cycloadditions of benzyne were significantly lower in energy than the [4+2] cycloadditions. These results suggested a possibility of separating different chiral and electronic types of CNTs using the reaction of aryne cycloaddition. The computed electronic band structures of graphene and CNTs functionalized with benzyne showed that cycloaddition reactions involving arynes could potentially be used for band gap engineering of graphene and CNTs.

Acknowledgement This research was supported by the National Science Foundation under Grant No. CHE1112388 and by a New Mexico State University Graduate Research Enhancement Grant.

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(2) Geim, A. K.; Novoselov, K. S. The Rise of Graphene. Nat. Mater. 2007, 6, 183−191. 47 48 49

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Solution Properties of Single-Walled Carbon Nanotubes. Science 1998, 282, 95−98. (4) Dai, L.; Mau, A. W. H. Controlled Synthesis and Modification of Carbon Nanotubes

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J. T. Microwave-Assisted Sidewall Functionalization of Single-Wall Carbon Nanotubes by Diels-Alder Cycloaddition Chem. Commun. 2004, 2004, 1734−1735.

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gin, N. G. Benzyne-Functionalized Graphene and Graphite Characterized by Raman Spectroscopy and Energy Dispersive X-ray Analysis. Carbon 2013, 54, 192−200.

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B864−B871. (25) Kohn, W.; Sham, L. J. Self-Consistent Equations Including Exchange and Correlation Effects. Phys. Rev. 1965, 140, A1133−A1138. (26) Ordejon, P.; Artacho, E.; Soler, J. M. Self-Consistent Order-N Density-Functional Calculations for Very Large Systems. Phys. Rev. B 1996, 53, R10441−R10444. (27) Soler, J. M.; Artacho, E.; Gale, J. D.; Garcia, A.; Junquera, J.; Ordejon, P.; SanchezPortal, D. J. The Siesta Method for Ab Initio Order-N Materials Simulation. J. Phys.: Condens. Matter 2002, 14, 2745−2779. (28) Kleinman, L.; Bylander, D. M. Efficacious Form for Model Pseudopotentials. Phys. Rev. Lett. 1982, 48, 1425−1428. (29) Troullier N.; Martins, J. L. Efficient Pseudopotentials for Plane-Wave Calculations. Phys. Rev. B 1991, 43 1993−2006. (30) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865−3868; Erratum ibid. 1997, 78, 1396. (31) Boukhvalov, D. W.; Katsnelson, M. I. Tuning the Gap in Bilayer Graphene Using

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Table 1: Binding energies, Eb , and equilibrium C−C bond lengths, deq , for graphene functionalized with benzyne. Calculations were carried out using 6 × 6 and 8 × 8 hexagonal graphene supercells.

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3

Attachment Position 1-2 1-3 1-4

Cycloaddition Reaction [2+2] [3+2] [4+2]

Graphene 6×6 + C6 H4 Eb (eV) deq (˚ A) −0.69 1.55 +0.45 1.57 −0.57 1.59

Table 2: Direct band gaps, Eg , at the Γ and K points and the minimum band gaps, Egmin , for graphene functionalized with benzyne in the 1-2 [2+2] and 1-4 [4+2] attachment positions. Calculations were carried out for n × n hexagonal graphene supercells with n = 3 − 9. The last line of the table shows the band gaps for graphene functionalized with two benzyne molecules per supercell. All values are in eV. Graphene Supercell 3×3 4×4 5×5 6×6 7×7 8×8 9×9 5 × 5 (2)

1-2 [2+2] Attachment 1-4 [4+2] Attachment Eg (Γ) Eg (K) Egmin Eg (Γ) Eg (K) Egmin 0.20 1.87 0.20 0.26 2.50 0.26 2.09 0.74 0 2.44 0.72 0 1.34 0.52 0 1.56 0.56 0 0.063 1.44 0.063 0.035 1.53 0.035 1.55 0.28 0 1.57 0.29 0 1.15 0.22 0 1.20 0.23 0 0.028 1.18 0.028 0.013 1.21 0.013 0.46 0.86 0 1.54 0.88 0

4 45 46 47 48 49 50 5 1 5 2 5 3 54 5 56 57 58 59 60

Graphene 8×8 + C6 H4 Eb (eV) deq (˚ A) −0.73 1.55 +0.44 1.57 −0.58 1.59



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Table 3: Binding energies, Eb , and equilibrium C−C bond lengths, deq , for (8,0) zigzag and (5,5) armchair CNTs functionalized with benzyne in different attachment positions. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3

Attachment Position 1-2 1-3 1-4 1-5 1-6 3-6

Cycloaddition Reaction [2+2] [3+2] [4+2] [3+2] [2+2] [4+2]

(8,0) CNT + C6 H4 (5,5) CNT + C6 H4 ˚ Eb (eV) deq (A) Eb (eV) deq (˚ A) −2.44 1.54 −2.37 1.49 −0.86 1.56 / 1.57 +0.21 1.57 / 1.59 +0.54 1.63 −1.52 1.58 +0.55 1.58 −1.46 1.55 −1.92 1.54 −1.95 1.54 −2.35 1.56 – –

Table 4: Binding energies, Eb , for semiconducting zigzag CNTs of different diameters, D, functionalized with benzyne in the most stable 1-2 [2+2] and 3-6 [4+2] attachment positions. (n,m)

D (˚ A)

(5,0) (7,0) (8,0) (11,0) (14,0) (16,0)

3.92 5.48 6.27 8.62 11.0 12.5

Eb (eV) 1-2 [2+2] 3-6 [4+2] −3.80 −3.93 −2.66 −2.77 −2.44 −2.35 −1.89 −1.83 −1.61 −1.55 −1.43 −1.48

Table 5: Binding energies, Eb , for metallic armchair CNTs of different diameters, D, functionalized with benzyne in the most stable 1-2 [2+2], 1-6 [2+2], and 1-4 [4+2] attachment positions. 4 45 46 47 48 49 50 5 1 5 2 5 3 54 5

(n,m)

D (˚ A)

(3,3) (4,4) (5,5) (8,8)

4.07 5.43 6.80 10.9

Eb (eV) 1-2 [2+2] 1-6 [2+2] 1-4 [4+2] −4.56 −3.11 −2.56 −3.10 −2.33 −1.88 −2.37 −1.95 −1.52 −1.32 −1.46 −1.08

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Figure 1: Surface functionalization of graphene and CNTs with benzyne. The numbers 1 through 6 show different attachment positions of benzyne molecules to (a) graphene, (b) zigzag CNTs, and (c) armchair CNTs. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3

Figure 2: Optimized structures of graphene sheets functionalized with benzyne in the 1-2 [2+2], 1-3 [3+2], and 1-4 [4+2] attachment positions. The graphene sheets modeled by the 6×6 hexagonal supercells are shown.

Figure 3: Band structures of (a,d) pristine graphene and graphene functionalized with benzyne in the (b,e) 1-2 [2+2] and (c,d) 1-4 [4+2] attachment positions. The electronic energy bands were computed using the 6×6 and 8×8 hexagonal graphene supercells. The zero energy corresponds to the Fermi level.

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Figure 4: Band structures of (a) pristine graphene and graphene functionalized with two benzyne molecules in the (b) 1-2 [2+2] and (c) 1-4 [4+2] attachment positions. The electronic energy bands were computed using the 5×5 hexagonal graphene supercell. The zero energy corresponds to the Fermi level. 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 4 0 4 1 4 2 4 3 4 45 46 47 48 49 50 5 1 5 2 5 3 54 5 56 57 58

Figure 5: Calculated electronic densities of states (DOS) of graphene functionalized with benzyne in the (a) 1-2 [2+2] and (b) 1-4 [4+2] attachment positions. The zero energy corresponds to the Fermi level.

Figure 6: Optimized structures of the (8,0) zigzag CNTs functionalized with benzyne in different attachment positions.

Figure 7: Optimized structures of the (5,5) armchair CNTs functionalized with benzyne in different attachment positions. The 3-6 [4+2] attachment configuration for the (5,5) CNT spontaneously converges into the 1-2 [2+2] attachment configuration.

Figure 8: Calculated band structures of the (a,b) (8,0) zigzag and (c,d) (5,5) armchair CNTs functionalized with benzyne in the [2+2] and [4+2] attachment positions. The energy bands of pristine and functionalized CNTs are shown by the dashed and solid lines, respectively. The zero energy corresponds to the Fermi level.

Figure 9: Calculated band gaps of the (a) (8,0) zigzag and (b) (5,5) armchair CNTs functionalized with benzyne in different attachment positions plotted as a function of the number of CNT unit cells. The dashed line corresponds to the calculated bend gap of the pristine (8,0) CNT.

Figure 10: Binding energies, Eb , for semiconducting zigzag CNTs functionalized with benzyne in the (a) 1-2 [2+2] and (b) 3-6 [4+2] attachment positions plotted as a function of the CNT inverse diameter, 1/D. The limit of D → ∞ corresponds to the binding energies of benzyne attached to graphene in the (a) 1-2 [2+2] and (b) 1-4 [4+2] positions. The best linear fits of Eb vs 1/D are shown by the dashed lines.

Figure 11: Binding energies, Eb , for metallic armchair CNTs functionalized with benzyne in the (a) 1-2 [2+2], (b) 1-6 [2+2], and (c) 1-4 [4+2] attachment positions plotted as a function of the CNT inverse diameter, 1/D. The limit of D → ∞ corresponds to the binding energies of benzyne attached to graphene in the (a)-(b) 1-2 [2+2] and (c) 1-4 [4+2] positions. The best linear fits of Eb vs 1/D are shown by the dashed lines.

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0.6

Band Gap (eV)

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

− 1−2 [2+2] − 1−6 [2+2]

0.4

− 1−4 [4+2]

0.2

0

4

5

6

7

Number of Cells


Page 35 of 37


1 2 3 4 5 6

(a) Zigzag CNT + C6 H4 [2+2] (1−2)

−3

(5,0)

−2

(7,0)

−1

(8,0)

Binding Energy (eV)

0

−4 −5 0

0.05

0.1 0.15 0.2 0.25 o−1 Inverse Diameter (A )

0.3

(b) 0

Zigzag CNT + C6 H4 [4+2] (3−6)

(5,0)

−3

(7,0)

−2

(8,0)

−1 (11,0)

Binding Energy (eV)

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6 0

(11,0)

9

(16,0) (14,0)

8

(16,0) (14,0)

7

−4 −5 0

0.05

0.1 0.15 0.2 o 0.25 Inverse Diameter (A−1)


0.3


Page 36 of 37

1 2 3 4 5 6

(a) Armchair CNT + C6 H 4 [2+2] (1−2)

−2 −3 −4 −5 0

0.05

(3,3)

−1

(4,4)

Binding Energy (eV)

0


0.3

(b) 0 Armchair CNT + C 6 H 4 [2+2] (1−6)

−3 −4 −5 0

0.05

(3,3)

−2

(4,4)

−1

(5,5)

Binding Energy (eV)

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6 0

(5,5)

9

(8,8)

8

(8,8)

7

0.1 0.15 0.2 o 0.25 Inverse Diameter (A−1)


0.3

Page 37 of 37


1 2 3 4 5 6 7 8 9

(c) 0 Armchair CNT + C 6 H 4 [4+2] (1−4)

(3,3)

−3

(4,4)

−2

(5,5)

−1 (8,8)

Binding Energy (eV)

10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 6 0

−4 −5 0

0.05



0.3

First-Principles Study of Graphene and Carbon Nanotubes

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