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Dimerization of C60 molecules within the single-walled carbon nanotube
Physics Letters A 327 (2004) 55–60 www.elsevier.com/locate/pla
Dimerization of C60 molecules within the single-walled carbon nanotube Fengquan Cui ∗ , Chuanfu Luo, Jinming Dong National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China Received 12 January 2004; received in revised form 13 April 2004; accepted 14 April 2004 Available online 28 April 2004 Communicated by R. Wu
Abstract The total potential energy of the C60 chains within the single walled carbon nanotubes (SWNTs) has been calculated by employing the van der Waals potential and tight binding molecular dynamics. The obtained results show that, under some conditions, the C60 molecules can be dimerized within (9, 9) tube, (16, 0) tube or some other tubes of larger diameters. The dimerization structure is shown to be more stable than the equally separated C60 structure. It is also found that the dimerization of C60 molecules depends on the space between the nanotube and the encapsulated C60 s, rather than on the nanotube chirality. 2004 Elsevier B.V. All rights reserved. PACS: 61.48.+c; 61.46.+w; 83.10.Rs Keywords: Dimerization; Nanotube peapod; Molecular dynamics
1. Introduction Since the discovery of single-walled carbon nanotube (SWNT) [1], a variety of carbon nanotubes have been synthesized. SWNTs are seamless cylinders made of a single graphite sheet, with diameters ranging from nearly 4 Å to more than 10 nm. The unique structure makes SWNTs more exciting materials as they represent the perfect, one-dimensional form
* Corresponding author.
E-mail addresses: [email protected] (F. Cui), [email protected] (J. Dong). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.04.028
of carbon, and also attracts considerable interest in exploring its practical application as multifunctional materials in nanoscaled gas sensor, switches, and molecular field effect transistors [2,3]. Due to hollow space of the tube, some atoms or molecules can be doped into the inner of SWNTs to engineer new nanostructural materials, called as carbon nanotube-peapods, which are useful in nanoelectronics, nanomechanics, and nanosensors, etc. It is considered that the incorporated fullerenes can alter or enhance the mechanical, transport, electrical and electronic properties of the nanotubes [4–6]. Through encapsulation of C60 into nanotube, the nanotube-peapod is formed with a self-assembled
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hybrid structure, which is a combination of onedimensional SWNTs with zero-dimensional fullerenes inside. The SWNT wall has an influence on the arrangement of C60 molecules, which may change the electronic-structure of the nanotube, depending on the space between the C60 and the tube. The first principles in local density approximation (LDA) calculations indicate that the encapsulation of C60 is exothermic or endothermic, depending on the size of the nanotube, and the energy gap of the peapod exhibits variation, depending on both the space and the fullerenes [7]. And there is charge transfer observed from C60 to the tube, which results in a drastic change in electron transport characteristics and electronic structure [8]. Now, although carbon nanotubepeapods can be synthesized in high yield from acidtreated SWNTs vacuum-annealed in the presence of added fullerens [9,10], the formation mechanism of the peapods has not been completely comprehended [11,12], and the structure of C60 chain inside the nanotube is not yet well known in the precision limit of the present experimental facilities. So, in order to understand the peapod properties, it is necessary to study the structures of the C60 molecule chain within the nanotubes. It is known that there are different phases formed for the solid C60 through cycloaddition reactions, among which is the polymerization of the molecules when the solid C60 is exposed to visible or ultraviolet light [13]. And coalescence of C60 molecules has been reported [14] and studied topologically [15]. They both are peculiar process for C60 molecules. Similarly, there are some high-resolution transmission electron microscopy (HRTEM) observations, indicating that C60 molecules within nanotube can be unequally separated [16–18]. It is found the distances between adjacent C60 units on chain within the nanotube are very different (from nearly 0.9 nm to more than 1.0 nm [17]). This raises a question: whether C60 molecules can be dimerized or polymerized inside the nanotube, which motivated us to study numerically in this Letter the possibility to dimerize the C60 chain in nanotube and its structural characters. In our study, we have made the tight binding molecular dynamics (TBMD) simulation of the dimerization and the total potential energy calculation to find the most stable structures.
2. Calculations In general, the diameter of the SWNTs in the peapods is about 1.3 nm to 1.4 nm, while the diameter of the C60 unit is about 0.71 nm, which indicates the nearest separation between the inserted C60 molecule and the nanotube wall is around van der Waals distance (∼ 0.3 nm). On the contrary, sometimes the separation between atoms on the adjacent different fullerenes is much smaller and sometimes is in the range of covalent bond. So, in our theoretical calculations, a combination of a recently developed van der Waals (VDW) potential [12] with a widely used tight-binding (TB) carbon–carbon potential [19–21] has been used to describe the peapod. The VDW potential is written as: V (r) =
C12 C6 − 6 r 12 r
(1)
where r is the carbon atom–atom distance, and the constants C12 , C6 are taken as 22 500 eV Å12 and 15.4 eV Å6 , respectively, which can be used to explain the binding energy of C60 and is in qualitative agreement with the experimental observations [12]. The TB potential has been used successfully to predict the structures, and electronic energy bands of crystalline linear carbon chains, graphite, diamond and other carbon clusters [20–22], and even used in calculation of solid C60 , in which it can yield good results for the structure, stability, and electronic property [24,25]. In the TB potential, the total energy (per atom) is expressed as: Etot{r1 , . . . , rN } = Ebs {r1 , . . . , rN } + U {r1 , . . . , rN }, (2) where ri is the coordinate of ith atom. The first part Ebs , “band-structure” energy (per atom), Ebs =
N 1 Ej N
(3)
j =1
is the sum of the eigenvalues over the occupied part of the electronic band structure, and Ej is calculated by the Slater–Koster empirical tight binding method [23]. The second part U {r1 , . . . , rN }, a short-ranged two-
F. Cui et al. / Physics Letters A 327 (2004) 55–60
Fig. 1. Configuration of two dimerized C60 molecules.
body potential, N 1 φ(rij ) U {r1 , . . . , rN } = 2N
(4)
i,j =1 i=j
is the sum of the ion–ion repulsion and the correction to the double counting of the electron–electron interaction in the band-structure energy Ebs , and φ(rij ) can be determined by first-principles total-energy and electronic-structure calculations. The details about the TB potential can be found in Ref. [19,20]. Since the photo-polymerization of solid C60 was reported, various arrangements of connected C60 units have been proposed and compared [26–29], among which, the dumb-bell-shaped molecule, formed by the cycloaddition reaction of double-bonds (the adjoining edges between two six-membered rings in C60 , also called 66 bonds in this Letter) between adjacent C60 units, is considered to be the most favorable isomer of the C60 dimer. Fig. 1 shows the structure of 66/66 (two double-bonds parallel) bonding between two adjacent C60 molecules. So, in our calculation, the initial structure of C60 chain is arranged with the 66 bonds parallel between two adjacent C60 molecules (Fig. 2).
3. Results and discussion Our calculations show that dimerization occurs only when the separation of two C60 units is in the range of 8.7–9.1 Å, which is close to that for the dimerization of the dimer (C60 )2 in solid C60 discussed above [13]. In our calculations, there is a local minimum energy corresponding to the structure of equally separated C60 units in a chain,but the dimerization structure is more favorable. A convincing example is that: when we initially let C60 units unequally separated, within 400 steps (the total energy
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difference is smaller than 10−6 eV) in our TBMD, the C60 molecules can be dimerized in the chain, i.e., the small separations become further shorted, but the big ones extend more. This also confirms the experimental observations about the variability in the separations of adjacent C60 molecules within the nanotube [16]. We have performed TBMD simulations on a chain of C60 molecules with several different initial configurations. When the C60 units are initially placed on a line (Fig. 2(a)) with equal-separation of 8.89 Å, which is a default initial distance for two adjacent C60 molecules in our calculations, and each carbon atom has small random velocity along the nanotube axis, they remain in the state with the equal distance even after more than 800 MD steps except that some intraseparations between the atoms change a little. However, when the initial random velocities (also on the direction of the nanotube axis) of carbon atoms in C60 molecules are chosen to be greater, they will evolve into the dimerization structure. And when the C60 units are placed in an unequal distance initially, they can also gradually evolve into the same dimerization structure within 500 MD steps for different periodicities of the C60 chain. The structure of a relaxed C60 chain is shown in Fig. 2(b). Now, the C60 @(10, 10) peapod is taken as an example, in which the interaction between the nanotube and C60 molecules is calculated by employing VDW potential because the inter-wall separation between (10, 10) nanotube (13.56 Å in diameter) and inside C60 is about 3.2 Å, which is in the range of the van der Walls distance. Due to the mismatch between the periodicity of nanotubes and that of C60 chains, there are energy modulations along the tube axis at different positions of the chain, and the length of nanotube is a little longer than that of the C60 chain. But the energy difference is so weak that it cannot affect the structure of dimerization. In our calculations, when the C60 separation on a line within (10, 10) tube is taken as 8.89 Å, the interaction energy modulation due to the different C60 positions along the tube axis is much less than the energy decrease due to the dimerization (0.001 eV versus 0.9 eV). In our calculations, we also find that the electronic band energy (per atom) decreases gradually and the elastic energy increases contrarily during the process of the dimerization. However, the energy decrease in electronic part is greater
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(a)
(b) Fig. 2. (a) The initial and (b) final configuration of C60 chain in the TBMD relaxation and the corresponding distances (Å).
Fig. 3. The interaction energy modulation between (17, 0) tube and a C60 molecule inside due to C60 ’s different positions along the (17, 0) tube axis.
than the increase in potential energy, and accordingly the total energy decreases steadily as the C60 chain is dimerized, which is similar to the Peierls transition. We also choose the C60 @(17, 0) peapod as a contrast. The interaction energy varies as Fig. 3, when a C60 unit moves along the axis of (17, 0) tube (13.30 Å in diameter). From Fig. 3, we can see that the maximal energy modulation due to different locations inside (17, 0) tube is less than 0.02 eV, but the energy decrease per C60 molecule due to the
dimerization is around 0.9 eV. So, the (17, 0) tube structure has little influence on the location of C60 units inside, and the C60 chain can be dimerized if its periodicity is in the range of 8.7–9.1 Å. From our calculations, we find that the tube chirality has little influence on the dimerization of C60 molecules in nanotube because dimerization of the C60 units plays a dominant role in energy decrease. And this conclusion is similar to that in Ref. [30], which made the reaction energy calculations on the encapsulation of the C60 s, and said that the reaction energy is independent of metallic or semiconducting characters of the nanotubes, but depends on the interwall spacing between the nanotube and the C60 molecules. To test the impact of the tube diameter on dimerization, we also studied some other tubes with different diameters. The results from our TBMD calculations show that, within (9, 9) tube, the C60 molecules favor the dimerization structure rather than the equally separated structure. The diameter (12.20 Å) of (9, 9) tube is remarkably smaller than that of (10, 10) tube, and it can be understood in the total-energy electronic structure calculations in which encapsulating C60 s into (9, 9) tube is theoretically found to be endothermic, but exothermic for (10, 10) tube [7]. So we calculate the interactions between C60 s and the nanotube by employing the TB potential, and also employing the VDW potential as a contrast. We find that C60 s
F. Cui et al. / Physics Letters A 327 (2004) 55–60
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Fig. 4. The configuration of C60 @(9, 9) peapod after relaxation.
will dimerize in (9, 9) tube (Fig. 4) even though the energy modulation along the tube axis is much larger than in (10, 10) tube. As to the (8, 8) tube, the nearest distance between the nanotube and the C60 s is in the range of covalent bonds, and there are sharp deformations for both (8, 8) tube and C60 molecules. The interaction between the nanotube and C60 s is so strong that the structure of (8, 8) tube is a decisive factor to determine the location and structures of C60 s inside, leading to no possibility of C60 s dimerization within (8, 8) tube. Similarly, the maximal energy modulation due to different locations inside (15, 0) tube (11.74 Å in diameter) is about 0.97 eV, but the energy decrease per C60 molecule due to the dimerization is only around 0.9 eV. So, the tube structure strongly influences the distribution of C60 units inside, accordingly preventing the dimerization from occurring. But (16, 0) tube (12.52 Å in diameter) cannot prevent C60 s inside from dimerizing for its wider space. Larger diameter tubes have even less impact on the structure of C60 chain, because adjacent C60 molecules have more strong interactions on each other than that between the tube wall and C60 s molecules. Now, we want to make a comparison of our results with those for the dimer (C60 )2 which is similar with our structure. In our calculations, when the dimerization happens, the longer and the shorter distances between C60 units are 9.2 and 8.6 Å, respectively. In contrast, the inter-molecule distance for a dimer (C60 )2 is 9.022 Å by the first-principles calculations [26]. In addition, in our calculations, the intermolecular bond lengths in the two closer neighboring C60 molecules are 1.592 Å and the other
two bonds (the intra-molecular bonds) in the fourmembered ring are 1.582 Å. But, for a dimer (C60 )2 they are 1.588 and 1.578 Å, respectively, obtained by the first-principles calculations [26]. From the comparison made above, it is clear that our results are reasonable. It has been confirmed that upon high temperature annealing, encapsulated C60 molecules may coalesce into interior tubes [14], and the coalescence process are proved to be plausible through atomic rearrangement called Stone–Wales bond rotations, which is energetically favorable and topologically transferable [15,31]. The result about dimerization has great implications for C60 molecules merging and fusion.
4. Conclusions In this Letter, we calculated the total potential energy of peapod C60 @(n, m) using tight binding potential model, and find that through dimerization, the C60 molecule chain has a manifest energy decrease. Our results show that, during the process of dimerization, the energy decrease in electronic part is greater than the increase in potential energy, which is similar to the Peierls transition. So we conclude that, if C60 molecules are pressed into the nanotube with high density, the C60 molecule chain can be dimerized within the (9, 9) tube, (16, 0) tube and some other larger diameter tubes. The space between the nanotube and C60 s inside, rather than the chirality of the tube, has a dicisive influence on the C60 s’ dimerization. And the result about dimerization structure has important im-
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plications for C60 molecules coalescence and merging within carbon nanotubes.
Acknowledgements This work was supported by the Natural Science Foundation of China under Grant No. 10074026 and No. A040108. The authors acknowledge also support from a Grant for State Key Program of China through Grant No. 1998061407.
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Dimerization of C60 molecules within the single-walled carbon nanotube Fengquan Cui ∗ , Chuanfu Luo, Jinming Dong National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, People’s Republic of China Received 12 January 2004; received in revised form 13 April 2004; accepted 14 April 2004 Available online 28 April 2004 Communicated by R. Wu
Abstract The total potential energy of the C60 chains within the single walled carbon nanotubes (SWNTs) has been calculated by employing the van der Waals potential and tight binding molecular dynamics. The obtained results show that, under some conditions, the C60 molecules can be dimerized within (9, 9) tube, (16, 0) tube or some other tubes of larger diameters. The dimerization structure is shown to be more stable than the equally separated C60 structure. It is also found that the dimerization of C60 molecules depends on the space between the nanotube and the encapsulated C60 s, rather than on the nanotube chirality. 2004 Elsevier B.V. All rights reserved. PACS: 61.48.+c; 61.46.+w; 83.10.Rs Keywords: Dimerization; Nanotube peapod; Molecular dynamics
1. Introduction Since the discovery of single-walled carbon nanotube (SWNT) [1], a variety of carbon nanotubes have been synthesized. SWNTs are seamless cylinders made of a single graphite sheet, with diameters ranging from nearly 4 Å to more than 10 nm. The unique structure makes SWNTs more exciting materials as they represent the perfect, one-dimensional form
* Corresponding author.
E-mail addresses: [email protected] (F. Cui), [email protected] (J. Dong). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.04.028
of carbon, and also attracts considerable interest in exploring its practical application as multifunctional materials in nanoscaled gas sensor, switches, and molecular field effect transistors [2,3]. Due to hollow space of the tube, some atoms or molecules can be doped into the inner of SWNTs to engineer new nanostructural materials, called as carbon nanotube-peapods, which are useful in nanoelectronics, nanomechanics, and nanosensors, etc. It is considered that the incorporated fullerenes can alter or enhance the mechanical, transport, electrical and electronic properties of the nanotubes [4–6]. Through encapsulation of C60 into nanotube, the nanotube-peapod is formed with a self-assembled
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hybrid structure, which is a combination of onedimensional SWNTs with zero-dimensional fullerenes inside. The SWNT wall has an influence on the arrangement of C60 molecules, which may change the electronic-structure of the nanotube, depending on the space between the C60 and the tube. The first principles in local density approximation (LDA) calculations indicate that the encapsulation of C60 is exothermic or endothermic, depending on the size of the nanotube, and the energy gap of the peapod exhibits variation, depending on both the space and the fullerenes [7]. And there is charge transfer observed from C60 to the tube, which results in a drastic change in electron transport characteristics and electronic structure [8]. Now, although carbon nanotubepeapods can be synthesized in high yield from acidtreated SWNTs vacuum-annealed in the presence of added fullerens [9,10], the formation mechanism of the peapods has not been completely comprehended [11,12], and the structure of C60 chain inside the nanotube is not yet well known in the precision limit of the present experimental facilities. So, in order to understand the peapod properties, it is necessary to study the structures of the C60 molecule chain within the nanotubes. It is known that there are different phases formed for the solid C60 through cycloaddition reactions, among which is the polymerization of the molecules when the solid C60 is exposed to visible or ultraviolet light [13]. And coalescence of C60 molecules has been reported [14] and studied topologically [15]. They both are peculiar process for C60 molecules. Similarly, there are some high-resolution transmission electron microscopy (HRTEM) observations, indicating that C60 molecules within nanotube can be unequally separated [16–18]. It is found the distances between adjacent C60 units on chain within the nanotube are very different (from nearly 0.9 nm to more than 1.0 nm [17]). This raises a question: whether C60 molecules can be dimerized or polymerized inside the nanotube, which motivated us to study numerically in this Letter the possibility to dimerize the C60 chain in nanotube and its structural characters. In our study, we have made the tight binding molecular dynamics (TBMD) simulation of the dimerization and the total potential energy calculation to find the most stable structures.
2. Calculations In general, the diameter of the SWNTs in the peapods is about 1.3 nm to 1.4 nm, while the diameter of the C60 unit is about 0.71 nm, which indicates the nearest separation between the inserted C60 molecule and the nanotube wall is around van der Waals distance (∼ 0.3 nm). On the contrary, sometimes the separation between atoms on the adjacent different fullerenes is much smaller and sometimes is in the range of covalent bond. So, in our theoretical calculations, a combination of a recently developed van der Waals (VDW) potential [12] with a widely used tight-binding (TB) carbon–carbon potential [19–21] has been used to describe the peapod. The VDW potential is written as: V (r) =
C12 C6 − 6 r 12 r
(1)
where r is the carbon atom–atom distance, and the constants C12 , C6 are taken as 22 500 eV Å12 and 15.4 eV Å6 , respectively, which can be used to explain the binding energy of C60 and is in qualitative agreement with the experimental observations [12]. The TB potential has been used successfully to predict the structures, and electronic energy bands of crystalline linear carbon chains, graphite, diamond and other carbon clusters [20–22], and even used in calculation of solid C60 , in which it can yield good results for the structure, stability, and electronic property [24,25]. In the TB potential, the total energy (per atom) is expressed as: Etot{r1 , . . . , rN } = Ebs {r1 , . . . , rN } + U {r1 , . . . , rN }, (2) where ri is the coordinate of ith atom. The first part Ebs , “band-structure” energy (per atom), Ebs =
N 1 Ej N
(3)
j =1
is the sum of the eigenvalues over the occupied part of the electronic band structure, and Ej is calculated by the Slater–Koster empirical tight binding method [23]. The second part U {r1 , . . . , rN }, a short-ranged two-
F. Cui et al. / Physics Letters A 327 (2004) 55–60
Fig. 1. Configuration of two dimerized C60 molecules.
body potential, N 1 φ(rij ) U {r1 , . . . , rN } = 2N
(4)
i,j =1 i=j
is the sum of the ion–ion repulsion and the correction to the double counting of the electron–electron interaction in the band-structure energy Ebs , and φ(rij ) can be determined by first-principles total-energy and electronic-structure calculations. The details about the TB potential can be found in Ref. [19,20]. Since the photo-polymerization of solid C60 was reported, various arrangements of connected C60 units have been proposed and compared [26–29], among which, the dumb-bell-shaped molecule, formed by the cycloaddition reaction of double-bonds (the adjoining edges between two six-membered rings in C60 , also called 66 bonds in this Letter) between adjacent C60 units, is considered to be the most favorable isomer of the C60 dimer. Fig. 1 shows the structure of 66/66 (two double-bonds parallel) bonding between two adjacent C60 molecules. So, in our calculation, the initial structure of C60 chain is arranged with the 66 bonds parallel between two adjacent C60 molecules (Fig. 2).
3. Results and discussion Our calculations show that dimerization occurs only when the separation of two C60 units is in the range of 8.7–9.1 Å, which is close to that for the dimerization of the dimer (C60 )2 in solid C60 discussed above [13]. In our calculations, there is a local minimum energy corresponding to the structure of equally separated C60 units in a chain,but the dimerization structure is more favorable. A convincing example is that: when we initially let C60 units unequally separated, within 400 steps (the total energy
57
difference is smaller than 10−6 eV) in our TBMD, the C60 molecules can be dimerized in the chain, i.e., the small separations become further shorted, but the big ones extend more. This also confirms the experimental observations about the variability in the separations of adjacent C60 molecules within the nanotube [16]. We have performed TBMD simulations on a chain of C60 molecules with several different initial configurations. When the C60 units are initially placed on a line (Fig. 2(a)) with equal-separation of 8.89 Å, which is a default initial distance for two adjacent C60 molecules in our calculations, and each carbon atom has small random velocity along the nanotube axis, they remain in the state with the equal distance even after more than 800 MD steps except that some intraseparations between the atoms change a little. However, when the initial random velocities (also on the direction of the nanotube axis) of carbon atoms in C60 molecules are chosen to be greater, they will evolve into the dimerization structure. And when the C60 units are placed in an unequal distance initially, they can also gradually evolve into the same dimerization structure within 500 MD steps for different periodicities of the C60 chain. The structure of a relaxed C60 chain is shown in Fig. 2(b). Now, the C60 @(10, 10) peapod is taken as an example, in which the interaction between the nanotube and C60 molecules is calculated by employing VDW potential because the inter-wall separation between (10, 10) nanotube (13.56 Å in diameter) and inside C60 is about 3.2 Å, which is in the range of the van der Walls distance. Due to the mismatch between the periodicity of nanotubes and that of C60 chains, there are energy modulations along the tube axis at different positions of the chain, and the length of nanotube is a little longer than that of the C60 chain. But the energy difference is so weak that it cannot affect the structure of dimerization. In our calculations, when the C60 separation on a line within (10, 10) tube is taken as 8.89 Å, the interaction energy modulation due to the different C60 positions along the tube axis is much less than the energy decrease due to the dimerization (0.001 eV versus 0.9 eV). In our calculations, we also find that the electronic band energy (per atom) decreases gradually and the elastic energy increases contrarily during the process of the dimerization. However, the energy decrease in electronic part is greater
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(a)
(b) Fig. 2. (a) The initial and (b) final configuration of C60 chain in the TBMD relaxation and the corresponding distances (Å).
Fig. 3. The interaction energy modulation between (17, 0) tube and a C60 molecule inside due to C60 ’s different positions along the (17, 0) tube axis.
than the increase in potential energy, and accordingly the total energy decreases steadily as the C60 chain is dimerized, which is similar to the Peierls transition. We also choose the C60 @(17, 0) peapod as a contrast. The interaction energy varies as Fig. 3, when a C60 unit moves along the axis of (17, 0) tube (13.30 Å in diameter). From Fig. 3, we can see that the maximal energy modulation due to different locations inside (17, 0) tube is less than 0.02 eV, but the energy decrease per C60 molecule due to the
dimerization is around 0.9 eV. So, the (17, 0) tube structure has little influence on the location of C60 units inside, and the C60 chain can be dimerized if its periodicity is in the range of 8.7–9.1 Å. From our calculations, we find that the tube chirality has little influence on the dimerization of C60 molecules in nanotube because dimerization of the C60 units plays a dominant role in energy decrease. And this conclusion is similar to that in Ref. [30], which made the reaction energy calculations on the encapsulation of the C60 s, and said that the reaction energy is independent of metallic or semiconducting characters of the nanotubes, but depends on the interwall spacing between the nanotube and the C60 molecules. To test the impact of the tube diameter on dimerization, we also studied some other tubes with different diameters. The results from our TBMD calculations show that, within (9, 9) tube, the C60 molecules favor the dimerization structure rather than the equally separated structure. The diameter (12.20 Å) of (9, 9) tube is remarkably smaller than that of (10, 10) tube, and it can be understood in the total-energy electronic structure calculations in which encapsulating C60 s into (9, 9) tube is theoretically found to be endothermic, but exothermic for (10, 10) tube [7]. So we calculate the interactions between C60 s and the nanotube by employing the TB potential, and also employing the VDW potential as a contrast. We find that C60 s
F. Cui et al. / Physics Letters A 327 (2004) 55–60
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Fig. 4. The configuration of C60 @(9, 9) peapod after relaxation.
will dimerize in (9, 9) tube (Fig. 4) even though the energy modulation along the tube axis is much larger than in (10, 10) tube. As to the (8, 8) tube, the nearest distance between the nanotube and the C60 s is in the range of covalent bonds, and there are sharp deformations for both (8, 8) tube and C60 molecules. The interaction between the nanotube and C60 s is so strong that the structure of (8, 8) tube is a decisive factor to determine the location and structures of C60 s inside, leading to no possibility of C60 s dimerization within (8, 8) tube. Similarly, the maximal energy modulation due to different locations inside (15, 0) tube (11.74 Å in diameter) is about 0.97 eV, but the energy decrease per C60 molecule due to the dimerization is only around 0.9 eV. So, the tube structure strongly influences the distribution of C60 units inside, accordingly preventing the dimerization from occurring. But (16, 0) tube (12.52 Å in diameter) cannot prevent C60 s inside from dimerizing for its wider space. Larger diameter tubes have even less impact on the structure of C60 chain, because adjacent C60 molecules have more strong interactions on each other than that between the tube wall and C60 s molecules. Now, we want to make a comparison of our results with those for the dimer (C60 )2 which is similar with our structure. In our calculations, when the dimerization happens, the longer and the shorter distances between C60 units are 9.2 and 8.6 Å, respectively. In contrast, the inter-molecule distance for a dimer (C60 )2 is 9.022 Å by the first-principles calculations [26]. In addition, in our calculations, the intermolecular bond lengths in the two closer neighboring C60 molecules are 1.592 Å and the other
two bonds (the intra-molecular bonds) in the fourmembered ring are 1.582 Å. But, for a dimer (C60 )2 they are 1.588 and 1.578 Å, respectively, obtained by the first-principles calculations [26]. From the comparison made above, it is clear that our results are reasonable. It has been confirmed that upon high temperature annealing, encapsulated C60 molecules may coalesce into interior tubes [14], and the coalescence process are proved to be plausible through atomic rearrangement called Stone–Wales bond rotations, which is energetically favorable and topologically transferable [15,31]. The result about dimerization has great implications for C60 molecules merging and fusion.
4. Conclusions In this Letter, we calculated the total potential energy of peapod C60 @(n, m) using tight binding potential model, and find that through dimerization, the C60 molecule chain has a manifest energy decrease. Our results show that, during the process of dimerization, the energy decrease in electronic part is greater than the increase in potential energy, which is similar to the Peierls transition. So we conclude that, if C60 molecules are pressed into the nanotube with high density, the C60 molecule chain can be dimerized within the (9, 9) tube, (16, 0) tube and some other larger diameter tubes. The space between the nanotube and C60 s inside, rather than the chirality of the tube, has a dicisive influence on the C60 s’ dimerization. And the result about dimerization structure has important im-
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plications for C60 molecules coalescence and merging within carbon nanotubes.
Acknowledgements This work was supported by the Natural Science Foundation of China under Grant No. 10074026 and No. A040108. The authors acknowledge also support from a Grant for State Key Program of China through Grant No. 1998061407.
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