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Controlling Cross Section of Carbon Nanotubes via Selective Hydrogenation Guangfen Wu,† Jinlan Wang,*,† Xiao Cheng Zeng,‡ Hong Hu,§ and Feng Ding*,§ Department of Physics, Southeast UniVersity, Nanjing 211189, People’s Republic of China, Department of Chemistry and Nebraska Center for Materials and Nanoscience, UniVersity of Nebraska, Lincoln, Nebraska 68588, and Institute of Textiles and Clothing, Hong Kong Polytechnic UniVersity, Kowloon, Hong Kong, People’s Republic of China ReceiVed: March 5, 2010; ReVised Manuscript ReceiVed: May 30, 2010
We systematically studied effects of selective hydrogenation of single-walled carbon nanotube (SWNT) on the shape of tube cross section based on a mechanical relaxation model and ab initio calculations. We found that fully hydrogenated SWNTs (FH-SWNTs) are energetically more favorable than partially hydrogenated ones. We uncovered a new channel for the strain relaxation at the nanoscale, in contrast to the known plasticity or buckling channel. We showed that the curvature strain energy of a cylindrical FH-SWNT can be significantly relieved by flipping a few rows of H atoms from outside to inside of the tube. We conclude that selective hydrogenation of SWNTs not only can be an effective way to achieve highly stable configurations of FHSWNTs but also can be used to control the shape of tube cross section (triangle, square, etc.) for nanomechanic applications. I. Introduction 2
3
The sp hybridized carbon is energetically very close to sp hybridized carbon, a reason why the cohesive energy of graphite is only slightly greater than that of diamond, and also the reason why enormously diverse hydrocarbon molecules can be formed. A recent surge in carbon nanostructure research and the potential application of carbon nanostructures in hydrogen storage have further stimulated the study of hydrogenation of sp2 carbon nanostructures, e.g., hydrogenation of carbon nanotube,1-8 graphite,9 graphene,10-15 and fullerene.16,17 Despite numerous experimental and theoretical efforts, there remain a number of fundamental questions unanswered regarding hydrogenation of carbon nanostructures. For example, what is the most optimal hydrogenation level for a hydrogenated SWNT (H-SWNT)? How does the curvature of a SWNT affect average binding energy of hydrogen and the formation of H-SWNT? Answers to these questions would be helpful to the design of FH-SWNTs for nanomechanical applications and for tube processing (e.g., selective etching of conducting SWNTs6). In this paper, we systematically studied energetic properties of various FH-SWNT configurations using both a simple mechanical model and ab initio calculations. We found a new channel for strain relaxation in addition to the conventional plasticity or buckling channel. The curvature energy of a cylindrical FH-SWNT can be significantly relieved through a chemical process by flipping a few rows of H atoms from outside to inside of the tube, leading to a break of the cylindrical symmetry. Such selective hydrogenation can be utilized to control polygonal cross section (e.g., triangle, square, etc.) of FH-SWNTs by design. A graphene is a two-dimensional (2D) single-layer sp2 carbon network, which can be viewed as a starting point to build many sp2 carbon nanostructures. For example, a SWNT can be viewed as a rolled-up graphene nanoribbon. It is known that hydrogena* To whom correspondence should be addressed. E-mail: jlwang@ seu.edu.cn and
[email protected]. † Southeast University. ‡ University of Nebraska. § Hong Kong Polytechnic University.
tion of a graphene on one side will build up strain in the graphene due to the sp3 C atom pyramidalization.18 As a consequence, buckling could occur if the internal strain is greater than a critical value. To maintain the flatness of the graphene and avert both the internal strain and buckling, an equal level of hydrogenation on both sides of the graphene is required.12 Previous theoretical calculations and modeling have shown that besides the internal strain, both the unpaired π electrons (radicals) and the interface between sp2 and sp3 carbon structures incur extra formation energy.12 Therefore, strain alleviation, chemical radical removal, as well as minimization of the interface between sp2 and sp3 carbon structures are three main requisites for achieving the most stable hydrogenated sp2 carbon network. It is known that the dense packing of H on both sides of the graphene can meet all three requisites and thus can stabilize the flat graphene.12 A fully hydrogenated graphene (FHG) with H equally packed on both sides (denoted by 50%|50%graphene) can be regarded as a new 2D hydrocarbon phase or as a supermolecule (i.e., the graphane11).10-12 As a rolled-up graphene nanoribbon, a SWNT entails extra curvature energy.19 A SWNT also can be viewed as a unique carbon nanostructure with a cylindrical/spiral symmetry.20 Mechanically, a SWNT is intrinsically different from bulk diamond in that the curvature energy of the tube wall cannot be relaxed. Similarly, rolling up a FH-G into a cylindrical FH-SWNT can result in extra curvature energy as well. Nevertheless, an H-SWNT can be modified with more degrees of freedom than a bare SWNT because H atoms can bind to the tube wall either inside or outside. Thus, by controlling the level of hydrogenation, outside and inside, one can alleviate the curvature energy with low compensation. For example, Yu and Liu have shown that hydrogenation of a graphene nanoribbon on one side rolls the ribbon into an H-nanotube.21 It is important to note that a naturally rolled-up H-nanotube carries no curvature energy, which is different from the case of a bare SWNT. II. Computational Method Ab initio calculations were carried out with gradient-correct density functional theory (DFT) with the Perdew-Burke-
10.1021/jp102005k 2010 American Chemical Society Published on Web 06/17/2010
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Figure 1. Average binding energy of chemisorbed hydrogen on a (20,0) SWNT as a function of the level of hydrogenation (defined by the ratio of number of H atoms to the number of C atoms). Insets show three typical configurations of hydrogenated SWNTs at different levels of hydrogenation.
Ernzerhof (PBE) exchange-correlation functional.22 All-electron double numerical polarized (DNP) basis sets that include polarized p(d)-valence orbitals for H(C), as implemented in the DMol3 package,23 were employed. Periodic boundary conditions were applied with at least 10 Å vacuum separations between SWCTs or graphene nanoribbons. The SWNT and graphene nanoribbons were fully relaxed with 10/6 k-points along the periodic direction, respectively, and the 2D graphene was relaxed with a mesh of 12 × 12 × 1 k-points. A convergence criterion of 10-6 au on total energy and electron density was adopted for the self-consistent field calculations. The global cutoff was set to 5.5 Å. Geometry optimization was performed by using the Broyden-Fletcher-Goldfarb-Shanno algorithm with a convergence criterion of 10-3 au on the displacement and the gradient, and 10-5 au on the total energy. III. Results and Discussions First, we focus on the optimal hydrogenation level for energetically preferred H-SWNTs. As in the case of hydrogenation of graphene, dense packing of H (full hydrogenation) is always necessary to completely eliminate chemical radicals and minimize the interfacial energy between sp2 and sp3 carbon structures. Thus, one expects that the most stable configuration should be a FH-SWNT. Indeed, this is supported by experimental measurements that 100% H-SWNTs are more stable than partially hydrogenated ones.7 Our DFT calculation also confirms this experimental fact. As can be seen from Figure 1, the V-shape relationship between average binding energy and level of hydrogenation clearly shows that the FH-SWNT is energetically the most favored. A large-diameter FH-SWNT is expected to behave just like a FH-graphene since its curvature energy, which is proportional to 1/D2 (D is the tube diameter), is negligible. So, a FH-SWNT with 50% H on each side of the tube wall (denoted by 50%|50%-SWNT) is expected to be energetically the most preferred configuration in the limit of D f ∞. In reality, however, diameters of SWNTs are typically within a range of 0.7-2.0 nm.19 As such, the curvature effect becomes very important. Theoretically, two other limiting cases of FH-SWNTs have to be considered, that is, 100%|0%-SWNT and 0%|100%-SWNT.2,24,25 The 0%|100%-configuration, in which the inside of the tube is fully hydrogenated, is always higher in energy than 50%|50%- and 100%|0%- configurations. Hence the 0%|100%-configuration is only favored for very thin FH-SWNTs.24,25 Next, we studied selective flipping of hydrogen atoms on a FH-SWNT. Obviously, random flipping H atoms can create local
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Figure 2. Relieving curvature strain energy of FH-SWNT by selective flipping of H rows from inside to outside of the tube (the flipped H rows are marked by red). (a f b) Flipping an H row from top to bottom of a FH-graphene leads a sharp change of FH-graphene tangent direction by an angle ∆. (c-g) FH-AM SWNT with 0, 1, 2, 4, and 8 flipped H rows, respectively.
strain.12 Also, flipping a circle of H along the tube circumference, from inside to outside or vice versa, creates inhomogeneity along the tube axial direction. Here, we suggest flipping H atoms in equally separated rows along the tube axial direction. In this way, the buildup strain in the SWNT can be significantly relaxed by changing the tube cross section from a circle to a lower symmetrical polygon. As a proof of principle, we show in Figure 2a,b that by flipping a row of chemisorbed H from the top of the FH-graphene to the bottom an abrupt change of graphene tangent direction can be induced by an angle ∆. On top of a flat FH-graphene, the flipped H row can be viewed as a line defect, which carries positive formation energy, γ. However, for a FH-SWNT, this formation energy is actually compensated by reduction of the curvature energy. One can see clearly from Figure 2c-f that flipping a few equally separated rows of H atoms from inside to outside of the tube reduces the overall curvature. In fact, for a FH-SWNT with nH equally separated and flipped H rows, the original circular cross section is divided into nH arcs. The total length of these arcs remains the same, i.e., 2πD, but the summation of their central angles becomes 2π - nH∆. Thus, the curvature of the cross section of FH-SWNT becomes
k ) 1/Deff ) 1/D[2π/(2π - nH∆)]
(1)
where Deff ) D[2π /(2π - nH∆)] is the effective diameter of these arcs. For an armchair (AM) FH-SWNT, ∆ ) 75.65°, and the tube-wall curvature energy, εc ≈ k2, is minimized at nH* ) [360/75.65] ) 5 (i.e., εc ≈ k2 f 0 at Deff f ∞), where [ ...] means rounding the number in the bracket to an integer. For the 100%|0%-SWNT configuration of an (n,n) AM FH-SWNT, nH* ) 2n. The tube wall becomes effectively flat when Deff f ∞ and the curvature goes to zero at n ) [360/151.3] ) 2, which is the case for an extremely thin nanotube (D ≈ 0.3 nm). Further increasing the tube diameter (or n) of the 100%|0%-SWNT leads to a negative curvature (k < 0) at which the curvature energy (which is proportional to k2) rises again. This explains why only very small FH-SWNTs prefer the 100%|0%-configuration. For the configuration of 0%|100%, nH ) -2n < 0, which always leads to rising curvature energy. Clearly, the three configurations of 50%|50%, 100%|0%, and 0%|100% are only special cases. In reality, there are many unexplored hydrogenation configurations besides the three, for instance, flipping nH ) [2π/∆] number of H rows can reduce the curvature energy more effectively. It is easy to show that
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Figure 3. H binding energy as a function of tube diameter and the number of flipped H rows (nH). Panels a, b (c, d) are 3D (2D) plots of AM and ZZ SWNTs, respectively. The 3D wireframes shown in panels a, b are plotted based on eq 4, using fitted parameters shown in Table 1.
flipping nH equally separated rows of H from inside to outside of a FH-SWNT (defined as a new configuration as nH-FHSWNT, nH is the number of flipped H rows) changes the overall energy by
∆E ) β(1/Deff2 - 1/D2)(πDlz) + γnHlz
(2)
where β/D2 is the curvature energy per unit area of a FH-SWNT, γ is the formation energy per length of a flipped H row, and lz is the tube length. On the other hand, a 50%|50%-SWNT can be viewed as a rolled-up 50%|50%-graphene (nH ) 0), which entails a curvature energy of β/D2. Compared to H adsorption in graphene, the curvature energy increases the overall energy of FH-SWNT. So the difference in binding energy of H between a 50%|50%-SWNT and a 50%|50%-graphene equals the difference in curvature energy between a FH-SWNT and the bare SWNT. Therefore, the binding energy of H in a 50%|50%SWNT is
εb ) εb0 + (R/D2 - β/D2)/F ) εb0 + (R - β)/(D2F)
(3) where εb0 is the binding energy of H in a 50%|50%-graphene, R/D2 is the curvature energy of a bare SWNT, and F ) 41 nm-2 is the atomic density of a graphene. In eq 3, we neglect the small change in diameter from a SWNT to a FH-SWNT (ab initio calculation shows that the change is less than 5%). The binding energy of H in an nH-FH-SWNT can be calculated from the equation:
εb(D,nH) ) εb0 + [R - β(2π - nH∆)2 /(4π2)]/(D2F) + γnH /(πDF) (4) In eq 4, εb0, R, β, γ, and ∆ are five parameters that can be used to determine the most favorable configuration for a hydrogenated carbon nanostructure. To validate eq 4, we performed a series of DFT calculations to determine these parameters and the binding energy for several nH-FH-SWNT configurations.
The binding energy of H on a FH-graphene is calculated by
εb0 ) [EG + NEH2 /2 - EFH-G]/N
(5)
where N is the number of H atoms in a unit cell, and EFH-G, EG, and EH2 are the energy of a fully hydrogenated graphene, a bare graphene, and an H2 molecule, respectively. The calculated value of εb0 ) 0.235 eV/H is in good agreement with previous results.11,12 The formation energy of a flipped H row, γ, and the angle, ∆, along AM or ZZ direction are calculated based on the model shown in Figure 2a,b. Here, γ is defined as the energy difference between two configurations shown in Figure 2, configurations a and b, respectively, and ∆ can be measured directly. Our calculation shows that γZZ (2.88 eV/nm) is significantly greater than γAM (1.38 eV/nm), and the angle ∆ZZ (33.63°) is notably less than ∆AM (76.59°), implying that a FHgraphene itself is highly anisotropic. Parameters R and β are obtained by fitting the curvature energy of a series of SWNTs and FH-SWNTs, respectively. Here we can see that β values (βAM ) 0.117 and βZZ ) 0.136 eV/C) are greater than R values (RAM ) 0.078 and RZZ ) 0.077 eV/C), indicating that a FH-graphene is more rigid than a bare one. This explains why the H binding energy of 50%|50%SWNT is reduced when the tube diameter becomes smaller (R - β < 0 in eq 3). Calculated H binding energies in various nH-FH-configurations with different nH, tube diameters, and chirality are displayed in Figure 3a,b. Fitting the bind energies to eq 4 gives another set of parameters (Table 1). It can be seen that the calculated binding energies from DFT are in excellent agreement with those from eq 4. Table 1 also shows that the differences between the fitted and calculated parameters are very small, which supports the proposed model. Now eq 4 can be used to determine energetically preferred configurations of AM and ZZ FH-SWNTs. As shown in Figure 3c, the 50%|50%-configuration (nH ) 0) is preferred for large AM SWNT (D > 2.7 nm). The configuration with nH ) 1, 2, 3, or 4 flipped H rows becomes more favorable for D in the range 2.7-2.2, 2.2-1.7, 1.7-1.1, and D < 1.1 nm, respectively. For FH-ZZ nanotubes, due to the large formation energy of γ, nH * 0 is only favored in a small diameter range (D < 0.91 nm) in
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TABLE 1: Calculated and Fitted Parameters in Eq 4 for Armchair (AM) and Zigzag (ZZ) SWNTs, respectivelya armchair
zigzag
parameters
fitted
DFT
fitted
DFT
εb0 (eV/H) R (eV) β (eV) ∆ (deg) γ (eV/nm)
0.238 0.081 0.107 68.80 0.389
0.235 0.078 0.117 76.59 0.339
0.242 0.096 0.136 32.15 1.329
0.235 0.077 0.136 33.63 1.227
a Details of calculation of the parameter are given in the text. The fitted parameters are obtained by fitting the data shown in Figure 3 to eq 4.
vacancy site) and subsequent H flipping over the CNT wall would occur due to the relatively low activation energy of E* ) 0.74 eV. At room temperature (T ≈ 300 K, kBT ) 24.5 meV, where T is the temperature and kB is the Boltzmann constant), the flipping rate of H can be estimated by using the equation f ) 1012 exp(-E*/kT) ≈ 1 s-1. The flipping rate would increase to f ) 104-105 at an elevated temperature, kBT ) 40 meV. Although polygonal cross sections of FH-SWNTs have not been observed in experiments yet, we believe that such an experiment is doable given that circular cross sections of bare SWNTs have been detected26 and a nearly 100% hydrogenation level has been achieved.7 IV. Conclusion
Figure 4. Lowest binding energy of H as a function of tube diameter. The energetically most preferred configurations are nH ) 4 for tubes smaller than (7,7) or nH ) 3 for tubes greater than (8,8) as shown in the insets. The binding energies of another two configurations (100%|0%and 50%|50%-) are plotted for comparison.
which nH ) 1-5 are the most preferred configurations for D in the range 0.91-0.83, 0.83-0.75, 0.75-0.67, and 0.67-0.60 nm, respectively. A significant difference between fully hydrogenated AM- and ZZ-SWNTs can be seen from Figure 3c,d. The binding energy of the most stable FH-AM-SWNT is notably greater than that of the most stable FH-ZZ-SWNT with the same diameter. For tubes with D ≈ 0.7 nm, the difference is ∼40% (∼0.30 eV per H in (5,5) SWNT and ∼0.18 eV per H in a (9,0) SWNT). Such a significant difference could be exploited for separation of mixedAM/ZZSWNTsbyusingahydrogenation-dehydrogenation cycle. Furthermore, the average binding energies in all favored configurations are within 0.18-0.4 eV per H for typical SWNTs with D > 0.6 nm. This binding-energy range is desirable for hydrogen storage at nonextreme condition. Figure 4 presents the average binding energy of H as a function of the tube diameter for FH-AM-SWNT, 100%|0%-SWNT, and 50%|50%SWNT in their favorable configuration. The binding energies for 100%|0%- and 50%|50%-SWNT show monotonically decreasing and increasing trends as a function of the tube diameter, respectively. A transition occurs at D ≈ 0.5 nm, below which the 100%|0%-SWNT configuration is more favored and above which the 50%|50%-SWNT is lower in energy. For the most optimal configurations shown in this work their energies change monotonically. Tubes smaller than (7,7) prefer a square-like cross section, while (8,8) and (9,9) tubes prefer triangular cross section. Besides thermodynamic stability of these polygonal FHSWNTs, the kinetics associated with flipping H atoms over the tube wall is also an important issue. Theoretically, on a 100%|0% hydrogenated (5,5) SWNT surface, the barrier for flipping a H atom from outside to inside is ∼1.51 eV.25 Hence, this flipping event is unlikely to occur near room temperature. But the barrier for flipping the next H near the site of the first one is significantly lowered to ∼0.74 eV. In real experiments, the flipping of the first H may occur at a defective site (e.g., a
We have shown that the mechanical curvature energy in a FH-SWNT can be relieved via flipping of equally separated H atomic rows from inside to outside of the tube wall. A simple mechanical relaxation model has been developed and verified by ab initio calculations. The interplay between the formation energy of flipped H rows and the relieved curvature energy results in a break of the cylindrical symmetry in FH-SWNTs with small diameters (i.e., < 2.7/0.91 nm for AM/ZZ SWNTs). FH-SWNTs with a polygonal cross section are predicted to exist and can be detected by using high-resolution tunneling electron microscopy (HRTEM). Also, hydrogenation can be used for identification or separation of SWNTs. Furthermore, regardless of the tube diameter (within a range) and chirality, the average binding energy of the H atom in SWNTs is in a desirable range for hydrogen storage at nonextreme condition. Hence, FHSWNTs are potentially useful as hydrogen storage media. This study offers a new insight into a different channel for strain energy relaxation at the nanometer scale, in addition to the conventional mechanical channels (e.g., buckling or plasticity). The simple mechanical model together with parametrization based on ab initio calculations can be extended to study other functionalized nanomaterials. Acknowledgment. This work is supported by the NSF (20873019), NBRP (2010CB923401), NCET (NCET-06-0470), SRFDP (20090092110025), and Peiyu Foundation of SEU in China (J.W.). F.D. is supported by PolyU internal research funds (No. 1-ZV3B, A-PD1U). X.C.Z. is supported by U.S. Office of Naval Research (N00014-05-1-0432) and the Nebraska Research Initiative. The authors thank the computational resource at Department of Physics, SEU. References and Notes (1) Dinadayalane, T. C.; Kaczmarek, A.; Lukaszewicz, J.; Leszczynski, J. J. Phys. Chem. C 2007, 111, 7376. (2) Bilic, A.; Gale, J. D. J. Phys. Chem. C 2008, 112, 12568. (3) Stojkovic, D.; Zhang, P.; Lammert, P. E.; Crespi, V. H. Phys. ReV. B 2003, 68, 195406. (4) Zhang, G. Y.; Qi, P. F.; Wang, X. R.; Lu, Y. R.; Mann, D.; Li, X. L.; Dai, H. J. J. Am. Chem. Soc. 2006, 128, 6026. (5) Nikitin, A.; Ogasawara, H.; Mann, D.; Denecke, R.; Zhang, Z.; Dai, H.; Cho, K.; Nilsson, A. Phys. ReV. Lett. 2005, 95, 225507. (6) Zhang, G. Y.; Qi, P. F.; Wang, X. R.; Lu, Y. R.; Li, X. L.; Tu, R.; Bangsaruntip, S.; Mann, D.; Zhang, L.; Dai, H. J. Science 2006, 314, 974. (7) Nikitin, A.; Li, X. L.; Zhang, Z. Y.; Ogasawara, H.; Dai, H. J.; Nilsson, A. Nano Lett. 2008, 8, 162. (8) Miller, G. P.; Kintigh, J.; Kim, E.; Weck, P. F.; Berber, S.; Tomanek, D. J. Am. Chem. Soc. 2008, 130, 2296. (9) Hornekaer, L.; Sljivancanin, Z.; Xu, W.; Otero, R.; Rauls, E.; Stensgaard, I.; Laegsgaard, E.; Hammer, B.; Besenbacher, F. Phys. ReV. Lett. 2006, 96, 156104. (10) Sluiter, M. H. F.; Kawazoe, Y. Phys. ReV. B 2003, 68, 085410. (11) Sofo, J. O.; Chaudhari, A. S.; Barber, G. D. Phys. ReV. B 2007, 75, 153401. (12) Lin, Y.; Ding, F.; Yakobson, B. I. Phys. ReV. B 2008, 78, 041402.
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