Structure and energetics of hydrogenated and dehydrogenated carbon tori

Carbon 43 (2005) 1165–1173 www.elsevier.com/locate/carbon

Structure and energetics of hydrogenated and dehydrogenated carbon tori O. Ponomarenko, M.W. Radny *, P.V. Smith School of Mathematical and Physical Sciences, The University of Newcastle, Callaghan 2308, Australia Received 16 August 2004; accepted 8 December 2004

Abstract It is shown that large-diameter, finite and extended, hydrogenated, single-walled carbon nanotubes are energetically and structurally unstable. The instability of short, large-diameter, finite hydrogenated tubes leads to the formation of hydrogenated circular carbon tori of small tubular cross-section. It is also shown that carbon toroidal structures, formed by desorption of hydrogen from the hydrogenated tori, are more energetically stable than the corresponding open-ended, finite single-walled carbon nanotubes. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Carbon nanotubes; Molecular simulation; Adsorption properties

1. Introduction Following the successful synthesis of multi-walled and single-walled carbon nanotubes [1], various modifications of these structures have been considered. In particular, it has been proposed that carbon toroidal structures may exhibit interesting electronic and magnetic properties depending upon their helicity, cross-section and circumference [2–6]. Ring-like bundles of single-walled carbon nanotubes (SWCNTs) with diameters between 300 and 500 nm and thicknesses ranging from 1.5 nm to 15 nm have been produced by the laser vaporisation method [7] and ultrasonic radiation from SWCNT ropes [8]. Rings of similar size were also made from ropes of double-walled [9] and multi-walled carbon nanotubes [10]. In recent experiments in which SWCNTs were chemically cut in buffer solution and then functionalised using bacterial proteins, small car* Corresponding author. Tel.: +61 2 49 215447; fax: +61 2 49 216907 (M.W. Radny). E-mail addresses: [email protected] (O. Ponomarenko), [email protected] (M.W. Radny).

0008-6223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2004.12.007

bon rings with external diameters of 18–35 nm were formed [11]. Rings can be either coils or tori. Carbon rings formed by laser vaporisation [7] show a seamless hexagonal network of carbon atoms on their shells. This type of ring is called a circular torus. It is believed that a circular torus is formed by bending a single nanotube and joining its two open ends via the formation of strong C–C covalent bonds [12]. Introduction of pentagonal–heptagonal (5– 7) defects into the shell of a SWCNT also changes the curvature of a tube and may lead to the formation of polygonal tori [13–16]. A coil structure is a tight solenoid-like spiral stabilised by strong shell–shell Van der Waals attractions. This type of ring was identified in the ultrasound-driven self-folding of SWCNT ropes [8]. It has been shown using elasticity theory that large circular tori are more energetically stable than polygonal tori. This theory also predicts a quadratic relationship between the radius of a SWCNT and the external radius of the circular torus that is formed by bending this nanotube and joining its ends [4]. It thus follows that the thinner the initial tube, the smaller the defect-free circular toroid that can be formed. The external diameters of

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O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

Fig. 2. Schematic of a section of a narrow, infinite hydrogenated single-walled carbon nanotube.

Fig. 1. Formation of a (4,0)60 torus from a finite nanotube: (a) by rolling the open ends of the short, large-diameter (30,30)4 nanotube inside the tube and then forming bonds between the two edges, (b) by bending the long narrow, open-ended (4,0)60 nanotube around and connecting its ends.

experimentally formed carbon rings [11], however, are significantly less than the values estimated from the elas-

ticity model [4,8,12]. Molecular mechanics studies have predicted the occurrence of small (100–2000 atoms) defect-free circular tori with thin elongated and circular cross-sections and suggested that these may be found as ‘‘possible graphenes among the graphite laser vaporisation products’’ [17]. Studies of the interaction of hydrogen with carbon nanotubes have shown that hydrogen atoms chemisorbed on the inner surface easily detach, forming molecular hydrogen inside the tube [18], while hydrogen chemisorption at the outer surface of the nanotube produces stable C–H configurations, for both low

Table 1 Structural data for CH tori with zig-zag cross-section Initial tube (m,n)k

Resulting torus (m,n)k

Diameter d0 (initial tube)

Diameter dt (tubular section of torus)

Diameter DT (external diameter of torus)

C–C bond length for torus (parallel to axis of tubular section)

C–C bond length for torus (not parallel to axis)

C–H bond length for torus

(40,40)3

(3,0)80

54.24

2.51

56.31

Inside 1.55 Outside 1.58

1.55 1.56

1.07 1.07

(30,30)4

(4,0)60

40.68

3.68

44.82

Inside 1.53 Outside 1.59

1.53 1.55

1.08 1.08

(24,24)5

(5,0)48

32.54

4.17

37.71

Inside 1.51 Outside 1.65

1.54 1.56

1.09 1.09

(20,20)6

(6,0)20*

27.12

4.71

32.35

Inside 1.33 Outside 1.66

1.52 1.56

1.08 1.10

All of the parameters are in Angstroms. Here (m,n) are the indices defining the helicity of the tubular shell, and k is an integer which specifies the number of hexagons along the axial direction of the tube. From simple algebraic considerations it follows that k ¼ N ð2m þ nÞ=ð4ðm2 þ mn þ n2 ÞÞ where N is the number of atoms in the carbon shell. The asterisk (*) denotes a CH torus with partially desorbed hydrogen atoms. The values denoted ‘‘inside’’ relate to the inner portion of the toroidal shell and those denoted ‘‘outside’’ to the outer portion of the toroidal shell.

O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

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Table 2 Structural data for CH tori with armchair cross-section Initial tube (m,n)k

Resulting torus (m,n)k

Diameter d0 (initial tube)

Diameter dt (tubular section of torus)

Diamter DT (external diameter of torus)

C–C bond length for torus (perpendicular to axis of tubular section)

C–C bond length for torus (not perpendicular to axis)

C–H bond length for torus

(60,0)4

(2,2)60

46.98

3.27

51.26

Inside 1.54 Outside 1.54

1.53 1.58

1.08 1.08

(40,0)6

(3,3)40

31.32

4.32

36.95

Inside 1.53 Outside 1.54

1.52 1.58

1.09 1.09

(30,0)8

(4,4)30*

23.48

6.65

27.99

Inside 1.40 Outside 1.52

1.39 1.58

1.07 1.10

The asterisk (*) denotes a CH torus with partially desorbed hydrogen atoms. All of the parameters are in Angstroms. The values denoted ‘‘inside’’ relate to the inner portion of the toroidal shell and those denoted ‘‘outside’’ to the outer portion of the toroidal shell.

(H:C 6 0.5) and high (H@C) chemisorption rates [19]. DFT-TB and DFT calculations have also shown that positioning the hydrogen on alternating sites on the inside and outside of the nanotubular surface might result in greater stability than chemisorption on just the outer surface of the tube [18,20]. It has also been predicted that the high curvature and high local strain at bent regions of nanotubes may enhance their chemical reactivity [21]. In this paper we discuss the atomic structure and energetics of small, hydrogenated and dehydrogenated, single-walled circular carbon tori that result from the structural and energetic instability of finite, short, hydrogenated SWCNTs. It is shown that the chemisorption of atomic hydrogen on the outer shell of finite, open-ended SWCNTs has a profound effect on the energetics and structural stability of these tubes leading, in some cases, to the formation of stable, hydrogenated circular tori. It has also been shown that the circular carbon tori that result from the removal of hydrogen atoms from the outer shell of optimised hydrogenated tori are more energetically stable than the corresponding finite, open ended SWCNTs.

The initial structures of the nanotubes were obtained by cutting a parallelogram from a graphene sheet and then rolling it up to form a tube. Topologically, a circular torus can be obtained either by rolling the open ends of a short, large-diameter, finite tube inside the tube and forming bonds between the atoms of the two edges, or bending a long, narrow, open-ended nanotube and joining its ends [12]. Schematics of these two ways of forming a circular torus from a finite, open-ended SWCNT are shown in Fig. 1a and b, respectively. The chemisorption of hydrogen on carbon nanotubes/tori has been modelled by creating a layer of hydrogen atoms bound on their outer surface, as illustrated in Fig. 2. All of the calculations have been carried out for finite carbon tubes and circular tori containing 480 carbon atoms. For the corresponding infinite tubes we have

2. Method and procedure The energetics and equilibrium geometries of hydrogenated carbon tubes and tori, and dehydrogenated carbon tori, were obtained from total energy minimisation calculations using the extended Brenner empirical potential [22,23]. The parametrisation of the hydrocarbon part of this potential is identical to that of the original Brenner method [22] which has been extensively used in recent simulations of carbon nanotubes [24,25], fullerenes [26,27] and carbon tori [28,29]. The calculations have been carried out using a classical molecular dynamics (MD) program employing the velocity form of the Verlet algorithm with the temperature set equal to zero [23].

Fig. 3. Structure of the unstable short, large-diameter (30,30)4 infinite CH nanotube.

O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

1.54 1.53 1.57 1.57 1.52 1.52 1.57 1.55

1.08 1.09 1.10 1.10 1.10 1.10 1.11 1.10

) (3 ,3

-2.3

(

zig-zag nano tubes

)

1.54 1.54 1.54 1.55 1.55 1.55 1.56 1.55

(8,0)

) 6,0

(5 ,0

(2,2)50 (3,3)40 (4,4)30 (5,5)24 (6,6)20 (8,8)15 (10,10)12 Average bond length

) (4,4

-2.1

(8,8) (10,0)

-2.5 -2.7 -2.9 -3.1

armchair nano tubes

(4, 0)

C–C (perpendicular C–C (not perpendicular C–H to tube axis) to tube axis)

(5,5)

-1.9

(2, 2)

Tube (m,n)k

-1.7

(3, 0)

Table 3 Structural data for infinite armchair CH tubes (all of the parameters are in Angstroms)

Binding energy, eV/atom

1168

2

4

6

(a)

8

10

12

diameter, Å

Table 4 Structural data for infinite zig-zag CH tubes (all of the parameters are in Angstroms) Tube (m,n)k (3,0)80 (4,0)60 (5,0)48 (6,0)40 (8,0)30 (10,0)24 (12,0)20 Average bond length

C–C (parallel to tube axis)

C–C (not parallel to tube axis)

C–H

1.57 1.56 1.54 1.60 1.52 1.52 1.51 1.55

1.56 1.55 1.54 1.55 1.55 1.56 1.56 1.55

1.07 1.08 1.09 1.09 1.10 1.10 1.10 1.09

(4,4) *

(6,0)*

-4500

(3,3)

-4550

(5,0)

infinite CH zig-zag nanotubes (4,0)

infinite CH armchair nanotubes

-4600 (2,2)

CH tori (zig-zag cross-section) CH tori (armchair cross-section)

-4650 (3,0)

-4700 3

4

5

*

-2.1 3) (3,

-2.3 -2.5

(6,0)*

(5,0)

-2.7

,2 (2

-2.9

)

armchair tubular cross-section zig-zag tubular cross section

(4,0)

(3,0)

-3.1 2

3

4

5

6

diameter, Å

Fig. 5. Average hydrogen binding energy of (a) infinite CH tubes, and (b) circular CH tori, as a function of the diameter of their tubular cross-section. Indices (n,0) and (n,n) define the helicity of the tubular cross-section of the tubes and tori. The asterisks (*) denote CH tori with partially desorbed hydrogen atoms.

-4450

Total energy, eV

4) (4,

-1.9

(b)

-4400

2

Binding energy, eV/atom

-1.7

6

7

8

diameter, Å Fig. 4. Total energy of CH circular tori and narrow, infinite nanotubes as a function of the diameter of their tubular cross-section. Indices (n,0) and (n,n) define the helicity of the tubular sections of the tori (filled symbols) and the corresponding narrow, infinite nanotubes (underlying empty symbols). The asterisks (*) denote CH tori with partially desorbed hydrogen atoms.

is computationally feasible for the molecular dynamic simulations and enables a sufficient number of tubes of armchair and zig-zag symmetry of different radius to be generated. As the number of atoms in these tubes has been kept constant, the length and diameter of the finite tubes are mutually dependent parameters, and hence only a limited number of tubes with different length, diameter, and helicities can be formed. The possible tubular/toroidal structures comprising 480 atoms are listed in Tables 1 and 2. To calculate the energies of the infinite and finite pure carbon tubes we have employed the energetic model developed in a previous paper [30].

3. Results and discussion 3.1. Structure and energetics of infinite, hydrogenated single-walled carbon nanotubes

employed a periodic unit cell which also contains 480 atoms. In the case of hydrogenated carbon tubes and tori (CH) the respective structures were composed of 480 C atoms and 480 H atoms. This number of atoms

We have found that only relatively narrow, hydrogenated, infinite SWCNTs preserve a perfect tubular shape with an undistorted hexagonal network of carbon atoms

O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

and C–H bonds distributed evenly along the radial directions, as shown schematically in Fig. 2. Optimisation of large-diameter, infinite, hydrogenated SWCNTs led to structural instability in the form of unevenly distorted, and chaotically directed or broken C–C and C–H bonds, as illustrated in Fig. 3. These observations agree with the density functional theory (DFT) calculations reported by Yildirim et al. [31] where it was shown that infinite CH nanotubes were stable for tubular radii ˚ . Analogous behaviour has been obsmaller than 6.25 A served for hydrogenated single-walled silicon nanotubes [32,33]. In both cases, the observed structural instability of the large-diameter nanotubes can be explained by the fact that the C–C–H (Si–Si–H) bond angles are made to deviate more significantly from the tetrahedral sp3 angle of 109.5° with increasing diameter of the hydrogenated tube. The distortion of hydrogenated tubes due to the preference of hydrogen atoms to form tetrahedral bonds with the carbon tubular shell has also been seen in the MD simulations of Volpe and Cleri [19] for (15,0) and (6,6) infinite SWCNTs with a low coverage of hydrogen (H:C 6 0.5) on the outer shell. The predicted bond lengths for the structurally stable, narrow infinite CH tubes are given in Tables 3 and 4, for armchair and zig-zag helicities, respectively. The average C–C bond length for both armchair and zig-zag nano-

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˚ . This is longer than for tubes was found to be 1.55 A the corresponding dehydrogenated SWCNTs. The aver˚ for age C–H bond length was determined to be 1.09 A ˚ zig-zag tubes and 1.10 A for the armchair tubes. These results are in good agreement with the structural data obtained for selected infinite CH tubes from ab initio DFT calculations [18,31]. Previous work has shown that the total energy of dehydrogenated infinite SWCNTs exhibits a 1/D2 dependence, where D is the diameter of the tube. This energy variation arises from the elastic strain energy of the cylindrically curved tubular shell. In contrast to this we have found that the total energy of narrow, infinite hydrogenated SWCNTs increases with increasing diameter of the tube as shown in Fig. 4. We have also calculated the average hydrogen binding energy EBH, defined as EBH ¼ ðECH  EC  N H EH Þ=N H ;

ð1Þ

where NH is the number of hydrogen atoms chemisorbed on the SWCNT, ECH and EC are the total energies of the hydrogenated SWCNT and corresponding pure SWCNT, respectively, and EH is the energy of a hydrogen atom. Plots of this average hydrogen binding energy, EBH, as a function of the initial diameter of both armchair and zig-zag tubes are shown in Fig. 5a. It

Fig. 6. Imperfect CH circular tori formed by rolling up (a) (30,0)8, (b) (20,20)6, and (c) (24,0)12 nanotubes.

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O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

Strain energy per C-H pair, eV/atom

0.12 (4,4)*

0.1 CH tori (armchair cross-section) 0.08

CH tori (zig-zag cross-section)

0.06

(5,0)

(3,3)

0.04 (6,0)*

0.02 (4,0) (3,0)

0 25

(2,2)

30

35

40

45

50

55

60

external diameter DT, Å Fig. 7. Strain energy per C–H pair of circular CH tori as a function of their external diameter DT. Indices (n,0) and (n,n) define the helicity of the tubular cross-section of the tori. The asterisks (*) denote the CH tori with partially desorbed hydrogen atoms.

should be noted that according to the definition given in Eq. (1), the binding energy will be negative if the CH system is stable. The average hydrogen binding energy for infinite CH tubes is seen to decrease in magnitude with the increasing diameter of the tube. Similar results for EBH for infinite CH nanotubes were obtained from the ab initio DFT calculations of Yildirim et al. [31]. While the results presented by Yildirim et al. suggest that zig-zag CH tubes have binding energies that are about 30 meV/atom smaller in magnitude than armchair CH tubes, our data does not show any systematic difference in EBH between armchair and zig-zag tubes. The EBH binding energies reported by Yildirim et al. [31] are also greater in magnitude than the values predicted by our empirical calculations (by 0.85 eV/atom for a (5,5) nanotube). 3.2. Energetics and relative stability of finite CH nanotubes and tori The instability of large-diameter, infinite CH tubes is reflected in the observed structural transformations of

Table 5 Structural data for dehydrogenated C tori with zig-zag cross-section Initial tube (m,n)k

Resulting torus (m,n)k

Diameter d0 (initial tube)

Diameter dt (tubular section of torus)

Diameter DT (external diameter of torus)

C–C bond length for torus (parallel to axis of tubular section)

C–C bond length for torus (not parallel to axis of tubular section)

(40,40)3

(3,0)80

54.24

2.21

56.85

Inside 1.41 Outside 1.45

1.49 1.47

(30,30)4

(4,0)60

40.68

3.29

43.57

Inside 1.48 Outside 1.43

1.38 1.46

(24,24)5

(5,0)48

32.54

3.70

36.39

Inside 1.35 Outside 1.52

1.41 1.47

(20,20)6

(6,0)20

27.12

4.16

31.20

Inside 1.34 Outside 1.55

1.40 1.48

All of the parameters are in Angstroms. The values denoted ‘‘inside’’ relate to the inner portion of the toroidal shell and those denoted ‘‘outside’’ to the outer portion of the toroidal shell.

Table 6 Structural data for dehydrogenated C tori with armchair cross-section Initial tube (m,n)k

Resulting torus (m,n)k

Diameter d0 (initial tube)

Diameter dt (tubular section of torus)

Diameter DT (external diameter of torus)

C–C bond length for torus (perpendicular to axis of tubular section)

C–C bond length for torus (not perpendicular to axis)

(60,0)4

(2,2)60

46.98

2.79

51.07

Inside 1.40 Outside 1.48

1.46 1.46

(40,0)6

(3,3)40

31.32

3.48

35.55

Inside 1.41 Outside 1.43

1.37 1.43

(30,0)8

(4,4)30

23.48

2.78 (corner) 4.65 (edge)

28.00*

Inside 1.30 (corner), 1.43 (edge) Outside 1.40 (corner), 1.44 (edge)

1.29 (corner), 1.32 (edge) 1.37 (corner), 1.40 (edge)

All of the parameters are in Angstroms. The values denoted ‘‘inside’’ relate to the inner portion of the toroidal shell and those denoted ‘‘outside’’ to the outer portion of the toroidal shell. The labels ‘‘corner’’ and ‘‘edge’’ used to describe the (4,4)30 torus are defined in Fig. 8b. *For this deformed torus, the stated external diameter is an average value.

O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

the corresponding finite CH tubes. We have found that optimisation of finite CH nanotubes leads to spontaneous bending of the open edges inside the tube (see also [33]). For very short, large-diameter CH tubes this bending results in carbon–carbon bonds being formed between the C atoms on the two edges of the tube and the formation of a circular torus as illustrated in Fig. 1a. It is interesting to note that tori formed from finite armchair nanotubes have zig-zag tubular cross-section, and vice versa. The various structural parameters that we have obtained from studying these finite, large-diameter CH tubes, and their resulting circular tori, are given in Tables 1 and 2. We have found that circular CH tori, with 480 carbon atoms on the shell, can be formed from ˚ finite CH tubes with diameters in the range of 24–47 A ˚ and 27–54 A for zig-zag and armchair tubes, respectively. The initial length of these tubes was found to ˚ for zig-zag tubes and 7.4– be in the range 8.5–17.5 A ˚ 14.8 A for armchair tubes. CH tori with large tubular cross-section and small external diameter DT are not structurally stable as illustrated in Fig. 6.

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The circular CH tori discussed above can also be formed by bending long, narrow, finite CH nanotubes and joining their open ends, as shown in Fig. 1b. The total energies of the tori formed from narrow finite CH tubes are also plotted in Fig. 4 as a function of the tubular diameter. The total energies of these CH tori are observed to increase with increasing diameter of the tubular cross-section, analogous to the infinite CH tubes, reflecting again the relative structural instability of large-diameter hydrogenated tubes and toroids. The results of calculations of the strain energies of our tori, defined as the difference between the total energy per C–H pair of a torus and its corresponding infinite CH tube, are shown in Fig. 7. The strain energy of our CH tori with both zig-zag and armchair symmetry of the tubular cross-section is seen to increase as the external diameter DT of the torus decreases. This explains the differences in the total energies of narrow, infinite CH tubes and CH circular tori seen in Fig. 4. The average hydrogen binding energies for CH tori were also calculated using Eq. (1) with ECH now being

Fig. 8. Two dehydrogenated carbon toroids: (a) the stable C torus which results from rolling up a (20,20)6 SWCNT, and (b) the optimised geometry of the (4,4)30 C torus obtained by rolling up a (30,0)8 SWCNT.

O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

the total energy of the CH torus, and EC the total energy of the corresponding dehydrogenated circular carbon torus. The results are plotted in Fig. 5b as a function of the diameters of the tubular cross-sections of the tori. The average hydrogen binding energy for CH tori is observed to decrease in magnitude with increasing diameter of the tubular cross-section, analogous to the behaviour of the infinite CH nanotubes shown in Fig. 5a.

-3200

Total energy, eV

1172

-3250

finite zig-zag C tubes

-3300

infinite zig-zag C tubes

-3350

armchair C tori (40,0) {3,3}

-3400 -3450 (30,0) {4,4}

-3500

The effects of the chemisorption of hydrogen on carbon tori can be further elucidated by comparison with the structure and energetics of small, dehydrogenated circular carbon tori. These C tori were obtained by removing the hydrogen layer from the optimised structures of the CH tori. After desorption of H, the resulting carbon toroidal shells were re-optimised. Values for the helicities, the C–C bond lengths and diameters of the tubular cross-sections of the resulting toroids are given in Tables 5 and 6 for zig-zag and armchair tubular cross-sections, respectively. It is found that hydrogen desorption generally leads to a decrease in both the C– C bond lengths and external diameters (DT) of the C tori (the structural data for the corresponding CH tori are given in Tables 1 and 2). The dehydrogenated circular tori with zig-zag cross-section were also found to keep a perfect, undistorted toroidal structure (Fig. 8a), while the structural stability of the armchair toroids was found to be size dependent. For example, the (4,4) toroid, which has the largest initial curvature of the shell with 480 C atoms, acquires a flattened rectangular shape as shown in Fig. 8b. While the hexagonal network of carbon atoms on the shell is preserved in this structure, some of the bond lengths are different from those in the undistorted regions of the shell (see Table 6). It has been shown that the energetics of finite, openended SWCNTs can be modelled as a balance between the strain energy of the tubular shell and the chemical energy of the dangling bonds on the open ends of the tube [30]. The energetics of finite SWCNTs is thus strongly influenced by the excess chemical energy which is proportional to the number of dangling bonds on the open edges of the tubes, and hence to the diameters of the tubes. The total energies of infinite and finite SWCNTs calculated using the energetic model [30] are shown in Fig. 9. The total energies of dehydrogenated carbon tori, considered as formed by rolling inward the edges of short, large-diameter SWCNTs (see Fig. 1a) are also plotted in Fig. 9 as a function of the diameter of the initial finite SWCNT. The total energies of the toroidal structures are seen to be lower than those of the corresponding large-diameter, finite SWCNTs. A similar lowering of the energy

-3550 0

10

20

30

40

50

60

diameter, Å

(a) -3200

finite armchair C tubes

-3250

Total energy, eV

3.3. Structure and energetics of dehydrogenated carbon circular tori

(60,0) {2,2}

(40,40) {3,0}

infinite armchair C tubes

-3300

zig-zag C tori

-3350

(30,30) {4,0}

-3400

(20,20) {6,0}

(24,24) {5,0}

-3450 -3500 -3550

0

(b)

10

20

30

40

50

60

diameter, Å

Fig. 9. Total energy of finite and infinite pure SWCNTs as a function of the diameter of their tubular cross-section, and of C tori as a function of their external diameter DT for (a) armchair, and (b) zig-zag toroidal cross-section. Indices (n,0), and (n,n) define the helicity of the initial finite large-diameter tubes. The numbers in the curly brackets define the helicity of the tubular cross-sections of the C tori.

due to a reduction in the chemical energy has been reported by Ihara et al. [16] for small polygonal carbon tori.

4. Summary In this paper, the atomic structure and energetics of short, hydrogenated and dehydrogenated carbon nanotubes and small circular tori have been discussed. The obtained results demonstrate that hydrogenated, infinite, large-diameter SWCNTs are energetically and structurally unstable. It has also been shown that the instability of short, large-diameter, open-ended hydrogenated finite nanotubes leads to the spontaneous formation of circular CH tori. Dehydrogenation of these CH toroids produces ‘‘clean’’, small, circular carbon tori. These dehydrogenated carbon tori were found to be more energetically stable than the corresponding short, large-diameter, open-ended SWCNTs. To check the stability of these structures in ambient conditions, finite temperature molecular dynamic simulations would need to be carried out. The chemisorption/desorption

O. Ponomarenko et al. / Carbon 43 (2005) 1165–1173

of hydrogen on the carbon tubular shell has been found to play a critical role in the observed structural and energetic transitions.

Acknowledgement One of us (O.P.) would like to thank the University of Newcastle for financial support. We would also like to thank the AC3 supercomputing facility for processing time, and the Space Physics Group of the University of Newcastle for access to their SUN ULTRAHPC450 computer.

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