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J. Phys. Chem. C 2008, 112, 11122–11129
From Fulleranes and Icosahedral Diamondoids to Polyicosahedral Nanowires: Structural, Electronic, and Mechanical Characteristics Jukka T. Tanskanen, Mikko Linnolahti,* Antti J. Karttunen, and Tapani A. Pakkanen* UniVersity of Joensuu, Department of Chemistry, P.O. Box 111, FI-80101 Joensuu, Finland ReceiVed: December 20, 2007; ReVised Manuscript ReceiVed: April 17, 2008
Structural relationships between icosahedral hydrocarbon nanostructures and their one-dimensional counterparts have been derived. The structural, electronic, and mechanical properties of fulleranes and icosahedral diamondoids, together with the predicted perhydrogenated carbon nanotubes and polyicosahedral diamond nanowires, were determined by quantum chemical calculations. Generally, the strain energies of the predicted one-dimensional hydrocarbon nanostuctures are low, being of the same magnitude with the experimentally known dodecahedrane and conventional diamond nanowires. The studied polyicosahedral diamond nanowires have Young’s moduli higher than the conventional diamond nanowires, suggesting useful applications in nanomechanical design. Introduction fullerenes,1,2
Since the discovery of the the nanostructures of carbon have attracted scientists due to their special characteristics compared with the respective bulk crystals. Alongside the fullerenes, the pristine carbon nanotubes (CNTs)3 are representatives of sp2 hybridized carbon nanostructures. On the other hand, higher diamondoids such as decamantane,4 or fully hydrogenated diamonds,5 belong to sp3 hybridized carbon nanostructures. Diamond-based materials have appealing properties, such as high elastic modulus and high strength-to-weight ratio. As a consequence, they have been suggested as optimal choices for nanomechanical designs.6 Accordingly, several experimental studies have focused on one-dimensional (1-D) diamond nanostructures such as carbon nanowires,7 nanowhiskers,8 diamond nanocylinders,9 and nanorods of single crystalline diamond.10 The different naming conventions, i.e., carbon nanowires, diamond nanowires (DNWs), and nanorods (DNRs), occasionally have been used interchangeably in the literature. Consistent naming conventions have been suggested, where the term “carbon” refers to the amorphous sp2/sp3 bonded structures while “diamond” refers exclusively to sp3 bonded structures. The terms “nanowire” and “nanorod” have been used in conjunction with infinite and finite 1-D carbon nanostructures, respectively.11 Several theoretical studies on diamond at the nanoscale have been reported, providing insight into the unique properties of the 1-D carbon nanostructures.12–15 Dehydrogenated DNWs have been shown to undergo reconstruction to form bucky-wires whereas hydrogenated nanowires are likely to preserve diamond surface morphologies.11,15 The electronic characteristics of both dehydrogenated and hydrogenated conventional 1-D diamond nanostructures have also been addressed.16 Generally, the theoretical studies on 1-D sp3 hybridized carbon nanostuctures have focused on conventional diamond structures, i.e., 1-D nanostructures superimposable with bulk diamond. The recently predicted icosahedral fulleranes17 and diamondoids18 represent sp3 hybridized carbon nanostructures not * To whom correspondence should be addressed. E-mail:
[email protected],
[email protected]. Fax: (+358) 13-251-3344.
superimposable with bulk diamond. Interestingly, oligomerization of the icosahedral fulleranes and diamondoids would produce polyicosahedral DNRs and DNWs not superimposable with bulk diamond. In the theoretical study reported herein, we focus on nonconventional diamond nanostructures, deriving the structural relationships between icosahedral cage hydrocarbons and their 1-D counterparts. The electronic, structural, and mechanical characteristics of the resulting 1-D hydrocarbon nanostructures will be discussed based on quantum chemical calculations. Comparisons will be made to conventional DNWs, which are superimposable with bulk diamond. Theory and Computational Details Molecular and periodic systems were studied by the hybrid density functional B3LYP method.19 Periodic systems were fully optimized by using the CRYSTAL06 quantum chemistry software.20 The program enables the utilization of line group symmetries21 for systems with 2-, 3-, 4-, and 6-fold rotational axes. The extra large integration grid, as implemented in the CRYSTAL06 code and the default optimization convergence threshold of 10-7 au, was applied in the calculations. In the SCF process the threshold for total energy was 10-8 au. The molecular calculations were performed by TURBOMOLE version 5.9.1.22 The structures of the molecules were optimized within their respective point group symmetries. To verify that symmetry constraints are not an issue, we also optimized a representative DNR, the polyicosahedral DNR, corresponding to two connected icosahedral C20@C80H60, in C1-symmetry. The optimization in C1-symmetry produced the same D5h-symmetric structure as the symmetry-constrained one. To enable comparisons between periodic and molecular calculations, a modified 6-21G* basis set adopted from the work of Catti et al.23 was used for carbon, the modified basis set being necessary for the periodic calculations. The standard 6-31G** basis set was used for hydrogen in both molecular and periodic calculations. To study the effect of thermodynamics, and to verify the structures as true local minima, vibrational frequencies were calculated for the smallest DNRs and DNWs. In the case of B3LYP calculated frequencies, a harmonic frequency scaling factor of 0.9614 was used.24 The 13C NMR spectra were
10.1021/jp7119262 CCC: $40.75 2008 American Chemical Society Published on Web 07/03/2008
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J. Phys. Chem. C, Vol. 112, No. 30, 2008 11123
calculated by TURBOMOLE25 and the NMR chemical shifts are given with respect to TMS. In the calculation of TMS, the SVP26 basis set was used for silicon. Strain Energies and Electronic Characteristics. The term “strain energy”, originating from the surface and bulk reconstructions of the studied hydrocarbons, is used to measure deviation from optimal sp3 hybridization. The strain energies of the hydrocarbon nanostructures were calculated by a methodology previously described for fulleranes and diamondoids.18 A short summary of the methodology is given below. The studied hydrocarbon nanostructures are composed of three types of sp3 hybridized carbons: (1) Those bound to four other carbons, i.e., core atoms (C unit), (2) those bound to three carbons and to one hydrogen (CH unit), and (3) those bound to two carbons and to two hydrogens (CH2 unit). For each hydrocarbon, the strain energy can be obtained according to eq 1.
∆E ) [E(Cx(CH)y(CH2)z) xE(C) - yE(CH) - zE(CH2)](x + y + z)-1 (1) where E(Cx(CH)y(CH2)z) represents the total energy of the optimized system, E(CH2) is the energy of the infinite polyethene chain per CH2 unit, E(CH) is the energy of the infinite hydrogenated diamond (111) slab per CH unit, and E(C) is the energy of the bulk diamond per C atom. The hydrogenated diamond (111) sheet and the infinite polyethene chain represent strain-free, unreconstructed surfaces with (CH)n and (CH2)n stoichiometries, respectively. Diamond, on the other hand, represents the strain-free, unreconstructed bulk structure for sp3 hybridized carbon.17,18,27 Accordingly, the strain energy can be considered to originate from the surface and bulk reconstructions of the hydrocarbons. Dangling bonds are not present in the studied systems and, hence, do not contribute to the strain energy. It should be noted that the equation has an advantage of being homodesmotic.28 The electronic properties of the studied structures were determined on the basis of B3LYP-calculated HOMO-LUMO gaps and band gaps for finite and infinite structures, respectively. Muscat et al.29 have shown that B3LYP reproduces experimentally observed gaps reliably for a wide variety of materials, including structures with sp3 hybridized carbon. Elastic Properties. Second-order derivatives of the total energy with respect to deformation along the principal axis of symmetry were numerically calculated for both equilibrium and distorted structures. The Young’s modulus (Y) is then determined from eq 2
Y)
( )
1 ∂2E V0 ∂ε2
σ)-
R - Req Reqε
(3)
where R is the radius at deformation and Req is the equilibrium radius ( ) 0). The shear modulus (G), describing the response of an object to shearing deformations, can be derived from the Poisson’s ratio and from the Young’s modulus (eq 4).
G)
Y 2(1 + σ)
(4)
Results and Discussion Structural Relationships between the Icosahedral Cage Hydrocarbons and Their One-Dimensional Counterparts. From the point of view of ring topology, the molecular structures of single-walled CNTs3 can be reproduced by suitably connecting two or more identical fullerenes.1,2 Being perhydrogenated fullerenes, the same applies for the fulleranes,17 suitable connection of which then produces perhydrogenated singlewalled CNTs. We considered the structural characteristics of the 1-D counterparts of the fulleranes, starting from the experimentally known C20H20 dodecahedrane.34 The studies were extended to the larger C80H80 and C180H180 fulleranes, previously shown to be thermodynamically favored over previously reported (CH)n cages, including the dodecahedrane.17 Combining two fulleranes into a nanotube is illustrated in Figure 1a for the cases of C80H80 and C180H180. The procedure applies likewise to any other pair of larger icosahedral fulleranes, as well as for derivation of longer nanotubes by combining three or more fulleranes. The larger fulleranes may be connected in various ways, the procedure shown for C80H80 and C180H180 in Figure 1a producing
(2) ε)0
where V0 is the equilibrium volume, E is the total energy of the system, and is the relative deformation along the principal axis. The described method for determination of Young’s modulus has been widely used for carbon nanostructures.30–32 The equilibrium volume can be determined by various techniques. For example, in the case of CNTs the volumes of hollow cylinders with shell thickness corresponding to either interlayer spacing in graphite or van der Waals radius of carbon have been adopted.32,33 We used the van der Waals volumes of the minimum energy structures as the equilibrium volumes. In periodic calculations, the van der Waals volumes of the unit cells were used as equilibrium volumes. The Poisson’s ratio (σ), describing the change in the shape of an object under stress, is obtained from eq 3
Figure 1. Combining fulleranes and icosahedral diamondoids into nanotubes and nanorods: (a) C80H80 (top) and C180H180 fulleranes (bottom) and (b) C20H20 (top), C20@C80H60 (middle), and C20@C80@C180H120 (bottom) icosahedral diamondoids. Dashed lines represent mirror planes (σ). Hydrogens are omitted for clarity.
11124 J. Phys. Chem. C, Vol. 112, No. 30, 2008 tubular structures analogous to the perhydrogenated (10,0) and (15,0) zigzag CNTs, respectively. The carbon frameworks of the icosahedral diamondoids are composed of covalently bound layers of the icosahedral fulleranes.18 For example, the molecular structure of the icosahedral C20@C80H60 diamondoid can be rationalized by replacing the endo-hydrogens of the C80H80 fullerane with carbons. Thereby, the carbon framework of dodecahedrane is generated inside the carbon framework of the C80H80 fullerane. We focused on the 1-D counterparts of the three smallest icosahedral diamondoids, i.e., C20H20, C20@C80H60, and C20@C80@C180H120. Note that the C20H20 dodecahedrane can be classified as both a fullerane and a diamondoid, whereas its infinite 1-D counterpart is a DNW. The procedure of connecting the icosahedral diamondoids is illustrated in Figure 1b, producing hydrogenated polyicosahedral DNRs. The analogous larger nanostructures can be derived by the same approach. The resulting carbon frameworks of the polyicosahedral DNRs are not superimposable with bulk diamond in contrast with the conventional DNRs. The structural characteristics of the described polyicosahedral nanowires are possibly extendable for heavier Group 14 elements and elemental hydrides. Previously, nonhydrogenated polyicosahedral silicon nanowires have been suggested as the lowest energy form of 1-D silicon.35 Alongside the novel hydrogenated polyicosahedral DNRs and DNWs, we studied the conventional DNWs for comparison, the latter being superimposable with bulk diamond, while the former are not. Hence, the conventional DNWs may be considered as 1-D counterparts of the conventional diamondoids.5 Three types of conventional DNWs were studied, namely those with the principal axes parallel to the [100], [110], and [111] directions of bulk diamond. Three fully hydrogenated structures of the same cross-sectional shape were considered for each orientation. Rectangular, rhomboid, and hexagonal cross-sections were studied for the [100], [110], and [111] DNWs, respectively. Notably, the cross-sectional shapes of the DNWs show resemblance with the synthesized DNRs.10 Icosahedral Fulleranes and Their 1-D Counterparts. Examples of the fulleranes combined to form nanotubes by the procedure described above (see Figure 1a) are illustrated in Figure 2. In addition to the icosahedral fulleranes and the corresponding infinitely long nanotubes, finite-length nanotubes were included in the study: three for the C80H80 fullerane, and two for C180H180. The B3LYP-calculated energetics, HOMOLUMO gaps, and band gaps are summarized in Table 1. The strain energies (∆E) of the 1-D counterparts of the larger fulleranes somewhat increase as a function of the tube length, indicating the fulleranes to prefer icosahedral structures. The increased strain is due to the fully exo-hydrogenated hexagons, carbon atoms of which are not capable of adopting optimal sp3 hybridization. While there are no fully exo-hydrogenated hexagons in the icosahedral fulleranes, their number gets larger as a function of the tube length. On the other hand, the relative proportion of the strained hexagons decreases as a function of the tube diameter. As a consequence, the strain energies of the 1-D counterparts of the fulleranes decrease as a function of the diameter of the tube. The short CNTs derived from the C180H180 fullerane are already energetically favored over the experimentally known dodecahedrane. On the basis of these data, it appears likely that the strain will be even lower for CNTs derived from larger icosahedral fulleranes, such as C320H320 and C500H500.18 The HOMO-LUMO gaps of the fulleranes and their 1-D counterparts are large, around 7.1-7.8 eV. The gaps decrease as a function of the length of the structure, approaching the
Tanskanen et al.
Figure 2. Icosahedral fulleranes (top left) and their 1-D counterparts (top right and bottom): (a) C80H80 and (b) C180H180.
TABLE 1: Dimensions, Strain Energies, and HOMO-LUMO/Band Gaps of the Icosahedral Fulleranes and Their 1-D Counterparts no. of gap combined diameter length stoichiometry cages (nm) (nm) ∆Ea ∆G298.15Ka (eV) C20H20 C35H30 C65H50 C125H90 C30H20
1 2 4 8 ∞
0.66 0.64 0.64 0.64 0.64
0.66 0.87 1.56 2.95 ∞
14.6 13.8 13.2 13.0 12.8
C80H80 C120H120 C200H200 C360H360 C80H80
1 2 4 8 ∞
1.09 1.09 1.09 1.09 1.08
1.09 1.46 2.34 4.10 ∞
13.5 16.3 19.3 21.3 23.8
7.75 7.64 7.33 7.13 7.03
C180H180 C270H270 C450H450 C180H180
1 2 4 ∞
1.51 1.50 1.50 1.51
1.51 2.16 3.48 ∞
9.8 11.7 13.1 15.5
7.65 7.41 7.23 7.13
13.9 14.5 15.0 15.3 16.2
8.49 8.26 8.14 8.05 8.01
a The strain energies in kJ mol-1 per C atom are calculated from eq 1.
value of the band gap calculated for the corresponding infinitely long system.
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J. Phys. Chem. C, Vol. 112, No. 30, 2008 11125 TABLE 2: Dimensions, Strain Energies, and HOMO-LUMO/Band Gaps of the Icosahedral Diamondoids and Their 1-D Counterparts no. of combined cages
diameter (nm)
length (nm)
∆Ea
gap (eV)
C20H20 C35H30 C65H50 C125H90 C30H20
1 2 4 8 ∞
0.66 0.64 0.64 0.64 0.64
0.66 0.87 1.56 2.95 ∞
14.6 13.8 13.2 13.0 12.8
8.49 8.26 8.14 8.05 8.01
C20@C80H60 C175H90 C325H150 C625H270 C150H60
1 2 4 8 ∞
1.09 1.09 1.09 1.09 1.09
1.09 1.59 2.56 5.04 ∞
8.3 8.7 9.0 9.1 9.4
7.39 7.26 7.14 7.11 7.13
C20@C80@C180H120 C490H180 C910H300 C280H120
1 2 4 ∞
1.52 1.53 1.53 1.53
1.52 2.31 3.92 ∞
6.3 7.0 7.5 8.2
7.01 6.81 6.68 6.65
C80H20@C180H120 C450H220 C830H380 C380H160
1 2 4 ∞
1.52 1.52 1.52 1.53
1.52 2.30 3.89 ∞
7.9 8.7 9.2 9.9
6.43 6.04 5.87 5.75
stoichiometry
a
Strain energies in kJ mol-1 per C atom are calculated from eq
1.
Figure 3. Icosahedral diamondoids (top left) and their polyicosahedral counterparts (top right and bottom): (a) C20H20, (b) C20@C80H60, and (c) C20@C80@C180H120.
Polyicosahedral Diamondoids. Molecular structures of three icosahedral diamondoids, C20H20, C20@C80H60, and C20@C80@C180H120, together with the corresponding polyicosahedral DNRs and DNWs are illustrated in Figure 3. In addition to the filled structures, we also studied the 1-D counterparts of a hollow two-layered C80H20@C180H120, which is structurally analogous to C20@C80@C180H120, the 20 core carbons being replaced by hydrogens. Similar to the case of icosahedral fulleranes, the icosahedral diamondoids, the corresponding infinitely long nanowires, and finite-length nanorods were included in the study. The B3LYP-calculated energetics, HOMO-LUMO gaps, and band gaps are summarized in Table 2. The strain energies suggest the 1-D structures to be favored over the dodecahedral C20H20. The low structural strain of the DNRs derived from the dodecahedrane is in agreement with previous studies on short oligomers of dodecahedrane.36,37 To take the effect of thermodynamics into account, we calculated the Gibbs corrected strain energies at T ) 298.15 K for
dodecahedrane and its 1-D counterparts. The Gibbs corrected strain energies were determined by eq 1 with all energy quantities replaced by the corresponding Gibbs free energies. The Gibbs corrections make the dodecahedrane slightly favored over its 1-D counterparts. The differences are small, however, suggesting the polyicosahedral DNRs and the DNW to be thermodynamically viable. With the exclusion of dodecahedrane, the strain energies increase, while only slightly, as a function of length of the polyicosahedral diamondoid, suggesting the preference for icosahedral structures. Combining the icosahedral diamondoids to polyicosahedral DNRs reduces the number of strain-inducing pentagons. However, connecting the cages introduces additional strain to the interface region due to fused pentagons. Nevertheless, the strain energies are systematically lower for the polyicosahedral DNRs than for the experimentally known dodecahedrane. The relative proportion of the fused pentagons becomes reduced as a function of the diameter, decreasing the strain energies of the polyicosahedral DNRs derived from the larger icosahedral diamondoids.18 Comparisons between the 1-D counterparts of C20@C80@C180H120 and the corresponding hollow C80H20@C180H120 show the filled structures to be energetically favored. The HOMO-LUMO gaps of the diamondoids and their 1-D counterparts are large, around 5.9-8.5 eV, the gaps being generally smaller for the structures with larger diameters. The HOMO-LUMO gaps decrease as a function of the length of the structure, approaching the value of the band gap calculated for the corresponding infinitely long DNW. Conventional Diamond Nanowires. Representative examples of the hydrogenated conventional DNWs, i.e., DNWs that are superimposable with bulk diamond, are illustrated in Figure 4. In this context, the focus of the studies was on periodic systems, the strain energies converging toward zero, i.e., toward the strain-free bulk diamond, as a function of the size of the
11126 J. Phys. Chem. C, Vol. 112, No. 30, 2008
Tanskanen et al.
Figure 5. Strain energies (∆E) as a function of the hydrogen-to-carbon ratio (H/C ratio) of the hydrogenated conventional and the polyicosahedral DNWs.
Figure 4. Front and side views of the hydrogenated conventional diamond nanowires having principal axes parallel to the (a) [100], (b) [110], and (c) [111] directions of bulk diamond.
TABLE 3: Dimensions, Strain Energies, and Band Gaps of the Hydrogenated Conventional Diamond Nanowires Having Principal Axes Parallel to the [100], [110], and [111] Directions of Bulk Diamond stoichiometry
orientation
diameter (nm)b
∆Ea
band gap (eV)
(C25H20)∞ (C49H28)∞ (C64H32)∞
[100] [100] [100]
0.85 1.21 1.39
6.0 4.7 4.3
6.81 6.40 6.32
(C18H12)∞ (C50H20)∞ (C72H24)∞
[110] [110] [110]
0.94 1.66 2.02
0.4 0.2 0.2
7.27 6.64 6.51
(C38H30)∞ (C74H42)∞ (C122H54)∞
[111] [111] [111]
0.79 1.08 1.38
4.8 4.1 3.5
7.35 6.66 6.37
a Strain energies in kJ mol-1 per C atom are calculated from eq 1. b Longest distance between hydrogens in plane perpendicular to the principal axis of symmetry.
system.18 The B3LYP-calculated energetics, HOMO-LUMO gaps, and band gaps of the conventional DNWs are summarized in Table 3. The calculated strain energies are clearly the lowest for those conventional DNWs that are parallel to the [110] direction of bulk diamond. The highest stability of the [110] DNWs can be understood to originate from surface hydrogenation. The H-H distances between the surface hydrogens are around 2.5 Å for the [110] DNWs, while they are down to 2.0 Å for the [100] and [111] DNWs. The repulsion between the surface hydrogens thus somewhat destabilizes the [100] and [111] DNWs, whereas the repulsion is negligible for the [110] DNWs. The described preference for hydrogenated [110] DNWs also has been experimentally observed, DNRs synthesized by hydrogen plasma post-treatment of multiwalled CNTs preferring the [110] growth direction.38 Previously, dehydrogenated DNWs have been shown
to prefer structures with principal axis parallel to the [100] direction, while [110] DNWs have been reported unstable.39 Due to the impact of H-H interactions at the surface, the presence (or absence) of hydrogen in the synthesis process of DNWs and DNRs may have an effect on the orientation of the products. Overall, the strain energies of the conventional [100] and [111] DNWs are of the same magnitude with the polyicosahedral DNWs. The strain energies of the DNWs can be related to the total surface-to-volume ratio, which can be approximated by the hydrogen-to-carbon ratio (H/C ratio), i.e., the number of hydrogens divided by the number of carbons. The strain energies of the conventional and the polyicosahedral DNWs are illustrated in Figure 5 as a function of the H/C ratio. The strain energies of both conventional and polyicosahedral DNWs decrease as a function of the diameter of the system, as the H/C ratio approaches zero. The strain is an order of magnitude lower for the conventional [110]. Nevertheless, the strain is actually low for each family of conventional DNWs, and corresponding DNRs are all experimentally known.10,38 On an energetic basis, this suggests the possible existence of the polyicosahedral DNWs, as well. The band gaps of the studied conventional DNWs are about the same as the band gaps calculated for the polyicosahedral DNWs (see Table 2). Mechanical Properties. The mechanical properties of the polyicosahedral and the conventional DNWs are of particular interest, the diamond structure having been suggested as an optimal choice for nanomechanical designs.6 Here we determine the mechanical properties, namely Poisson’s ratios, Young’s moduli, and shear moduli, for the studied hydrocarbon nanostructures. For a point of comparison, the elastic properties of a zigzag (22,0) CNT were determined by periodic calculations. To obtain the Young’s moduli for periodic systems, we both compressed and elongated the structure by varying the length of the periodic cell in the axial direction, followed by relaxation of the atomic positions. Likewise, the elastic moduli for the molecular systems were determined by distorting the equilibrium structures. The structurally distorted molecules were reoptimized under constraint of a specified and fixed length in the orientation of the principal axis of symmetry. Both periodic and molecular systems were distorted for (0.04 Å at intervals of 0.01 Å. The calculated mechanical properties are summarized in Table 4. The elastic moduli increase as a function of the length of the structure, as is clear from the calculated mechanical properties of C20H20 and C20@C80H60 cages and their finite 1-D counterparts. The Young’s moduli increase because the stress caused by distortion from the equilibrium becomes distributed among a larger number of C-C bonds. The increase in shear moduli is due to larger shear surface. The mechanical properties of finite structures converge toward the corresponding infinite systems. Therefore, periodic models were employed elsewhere.
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J. Phys. Chem. C, Vol. 112, No. 30, 2008 11127
TABLE 4: Young’s Moduli (Y), Shear Moduli (G), and Poisson’s Ratios (σ) of the Studied Hydrocarbon Nanostructures stoichiometry
combined diameter Y G cages/length (nm) (GPa) (GPa)
σ
C20H20 C35H30 C65H50 C125H90 C30H20
1 2 4 8 ∞
0.66 0.64 0.64 0.64 0.64
293.6 430.1 531.2 595.7 607.9
125.9 194.3 239.3 268.2 285.4
0.17 0.11 0.11 0.11 0.06
C80H80
∞
1.08
559.4
253.3
0.10
C180H180
∞
1.51
547.7
260.6
0.05
C80H20@C180H120
∞
1.53
711.6
298.1
0.19
C20@C80H60 C175H90 C325H150 C625H270 C150H60
1 2 4 8 ∞
1.09 1.09 1.09 1.09 1.09
266.4 359.9 468.6 568.9 694.0
122.7 163.0 206.4 245.2 296.8
0.09 0.10 0.14 0.16 0.17
C20@C80@C180H120
∞
1.53
750.0
309.0
0.21
[100] DNR [100] DNR [100] DNR
∞ ∞ ∞
0.85 1.21 1.39
367.2 419.9 448.0
163.6 181.8 197.1
0.12 0.16 0.14
[110] DNR [110] DNR [110] DNR
∞ ∞ ∞
0.94 1.66 2.02
607.3 660.6 673.4
272.7 298.6 305.0
0.11 0.11 0.10
[111] DNR [111] DNR [111] DNR
∞ ∞ ∞
0.79 1.08 1.38
528.2 601.7 678.0
252.5 286.5 323.9
0.05 0.05 0.05
(22,0) CNT
∞
1.72
897.7
386.0
0.16
Figure 6. Young’s moduli (Y) of the hydrogenated conventional and the polyicosahedral DNWs as a function of the cross-sectional area of the stucture.
TABLE 5: Infrared C-H Stretching Vibrations of the Conventional [100], [110], and [111] DNWs and the Polyicosahedral DNW Derived from C20H20 conventional DNWs [100] [110]
In general, the Young’s and shear moduli follow the same trends. The calculations give the highest Young’s modulus for the CNT, included as a reference, a value of about 900 GPa being in agreement with previous theoretical and experimental work.40–42 However, it should be noted that the flexible structures of the CNTs may reduce their applicability for nanomechanical designs.43,44 The diamond-like structure could be a more practical choice for applications where structural rigidity is critical. The studied conventional DNWs have Young’s moduli around 360-680 GPa, the moduli increasing as a function of the wire diameter. For approximately the same cross-sectional area of the wire, the [100] DNWs have clearly lower moduli than the [110] and [111] DNWs. At a cross-sectional area of 1.4 nm2 and beyond, the [111] DNWs have the highest Young’s moduli of the studied conventional DNWs (Figure 6). This in agreement with previous calculations, suggesting the [111] direction to have the highest Young’s modulus for the lowindex orientations of bulk diamond.13 With respect to its cross-sectional area, the Young’s modulus of the DNW derived from dodecahedrane is high, over 600 GPa. The rigidity of the 1-D dodecahedrane is associated with the presence of fused pentagons, which strongly resist further structural deformation under axial stress. The 1-D counterparts of dodecahedrane could form bundled structures, in a similar way that the CNTs form carbon nanoropes,45 and could have potential in mechanical applications, such as strengthening components in composite materials. The Young’s moduli of both hydrogenated conventional DNWs and the polyicosahedral DNWs are presented in Figure
[111] polyicosahedral DNW (C30H20)∞
wavenumber (cm-1)
assignment
2880-2910 2920-2940 2820-2850 2870-2900 2830-2880 2930-2980
CH CH2 CH CH2 CH CH2
2860-2910
CH
6 as a function of the cross-sectional area. Similar to the conventional DNWs, the elastic moduli of the polyicosahedral DNWs increase as a function of the cross-sectional area. Furthermore, their Young’s moduli are even higher than those of the conventional DNWs. Accordingly, the polyicosahedral DNWs, while being somewhat more strained than the conventional DNWs, could turn out to be valuable in nanomechanical designs. The perhydrogenated CNTs corresponding to C80H80 and C180H180 fulleranes have Young’s moduli considerably smaller than the modulus of the CNT. This is due to the stronger bonds between sp2 than sp3 hybridized carbons. The elastic moduli are also lower for the perhydrogenated CNTs than for the DNWs. Concerning the icosahedral C80H20@C180H120, which is an intermediate between the C180H180 fullerane and the C20@C80@C180H120 diamondoid, the elastic moduli of its infinite 1-D counterpart fits between the values calculated for the perhydrogenated CNT and the polyicosahedral DNW. Spectral Features. Infrared and NMR spectroscopy are useful for characterization of hydrocarbons. To find out if the polyicosahedral and the conventional diamond nanostructures could be distinguished with the spectroscopic tools, we determined their infrared and 13C NMR spectra. The infrared spectra were calculated for a representative of each conventional DNW, namely (C25H20)∞ for [100], (C18H12)∞ for [110], and (C38H30)∞ for [111], as well as for the polyicosahedral DNW derived from C20H20. The 13C NMR spectra were calculated for three polyicosahedral diamondoids composed of two icosahedral units (see Figure 1b), i.e., C35H30, C175H90, and C490H180. The calculated C-H stretching vibrations are summarized in Table 5. The conventional DNWs show C-H stretching vibrations associated with the CH and CH2 groups at wavenumbers around 2820-2980 cm-1. For the polyicosahedral DNW, the C-H stretching vibrations due to the CH groups occur at 2860-2910 cm-1, overlapping with those of the conventional DNWs. However, due to the absence of CH2
11128 J. Phys. Chem. C, Vol. 112, No. 30, 2008
Tanskanen et al. Conclusions The structural, electronic, and mechanical characteristics of novel hydrocarbon cages, nanotubes, and nanowires have been studied by both periodic and molecular B3LYP calculations. The icosahedral hydrocarbon cages, i.e., fulleranes and diamondoids, can be extended to produce polyicosahedral hydrocarbon nanostructures. In analogy to connecting fullerenes to carbon nanotubes, the fulleranes can be connected to produce perhydrogenated carbon nanotubes. Correspondingly, connecting the icosahedral diamondoids produces polyicosahedral diamond nanorods and nanowires. In contrast to the conventional diamond nanostructures, icosahedral diamondoids and their one-dimensional counterparts have carbon-frameworks that are not superimposable with bulk diamond. The strain energies of the perhydrogenated carbon nanotubes derived from the fulleranes generally somewhat increase as a function of the length of the nanotube. The same applies for the polyicosahedral diamond nanorods, making the icosahedral fulleranes and diamondoids energetically favored. Nevertheless, the strain energies of the perhydrogenated carbon nanotubes and the polyicosahedral diamond nanorods and nanowires are generally lower than those for the experimentally known dodecahedrane. Furthermore, the strain energies systematically decrease as a function of the diameters of the nanostructures. The strain energies of the predicted polyicosahedral diamond nanowires are of the same magnitude with the hydrogenated conventional [100] and [111] diamond nanowires, carbon frameworks of which are superimposable with bulk diamond. Infrared and 13C NMR spectroscopy could be used to characterize polyicosahedral diamond nanorods and nanowires. The polyicosahedral diamond nanowires possess useful mechanical properties. With respect to their diameter or crosssectional area, the Young’s moduli are higher for the polyicosahedral diamond nanowires than for the hydrogenated conventional diamond nanowires. As a consequence, the predicted polyicosahedral diamond nanowires could turn out to be useful in nanoscale mechanical design.
Figure 7. 13C NMR spectra of the conventional octahedral C84H6418 diamondoid and polyicosahedral DNRs derived from icosahedral C20H20, C20@C80H60, and C20@C80@C180H120 diamondoids.
groups, the spectrum of the polyicosahedral DNW has fewer absorptions than the spectra of the conventional DNWs. The spectra of both the conventional and the polyicosahedral DNWs show absorptions also at around 1200-1500 cm-1, which are due to C-H deformations of the CH and CH2 groups. Again, the polyicosahedral DNW has fewer absorptions than the conventional ones. The calculated 13C NMR spectra are shown in Figure 7, where the polyicosahedral DNRs are compared with the previously reported calculations on the conventional octahedral C84H64 diamondoid.18 In this context, the B3LYP/6-21G* level of theory has been shown to predict 13C chemical shifts of adamantane within 3 ppm.18 The 13C NMR spectra of the polyicosahedral DNRs are distinctly different from the spectrum of the conventional C84H64 diamondoid, which shows nine chemical shifts in the range of δ 45-33 ppm. The polyicosahedral DNR derived from the C20H20 has all chemical shifts above δ 45 ppm, and is thus clearly distinguishable from C84H64. The DNRs derived from C20@C80H60 and C20@C80@C180H120 both have six shifts above δ 45 ppm. The highest shifts decrease as a function of the diameter of the DNR. To conclude, we find that both infrared and, in particular, 13C NMR spectroscopy could assist in the characterization of the polyicosahedral DNRs.
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