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Lithium diffusion in single-walled carbon nanotubes: a theoretical study
Chemical Physics Letters 374 (2003) 548–555 www.elsevier.com/locate/cplett
Lithium diffusion in single-walled carbon nanotubes: a theoretical study Carolina Garau, Antonio Frontera *, David Qui~ nonero, Antoni Costa, Pau Ballester, Pere M. Deya 1 Departament de Quimica, Universitat de les Illes Balears, Crta. de Valldemossa km 7,5 07122 Palma de Mallorca, Spain Received 11 March 2003; in final form 11 April 2003
Abstract Theoretical investigations on single-walled carbon nanotubes (SWCNT) have been carried out using ab initio and molecular interaction potential with and without polarization (MIPp and MIP, respectively) methodologies to explore the possibilities of Liþ ion insertion through the side-wall of the nanotube. Li-nanotube systems can improve the capacity of lithium batteries by using both nanotube exteriors and interiors. Ab inito calculations of the fully optimized nanotubes were used to examine the topological defects depending on the ring size. The movement of the lithium cation through 8, 9 and 10-membered rings is studied and discussed. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Carbon nanotubes [1,2] have a wide range of structural, mechanical and electronic properties [3]. Mainly, they represent a new class of materials for exploring nanoelectronics and molecular electronics [4,5]. A single-walled carbon nanotubes (SWCNT) can be either metallic or semiconducting material [6], depending on its diameter and chirality. This unique ability to be both metallic and semiconducting without doping has led to speculation that SWCNT might serve as a key building block for the construction and design of carbon-based electronics [4,5]. For instance, one *
1
Corresponding author. Fax: +34-971-173426. E-mail address: [email protected] (A. Frontera). Also corresponding author.
promising direction for the transistors of the future involves molecular electronics in which the active part of the device is composed of a single or a few molecules [7]. A SWCNT can be viewed as a seamless cylinder obtained by rolling-up a section of a two-dimensional graphene sheet (Fig. 1). It is uniquely characterized by the roll-up vector C ¼ na1 þ ma2 ¼ ðn; mÞ, where a1 and a2 are the primitive lattice vectors of the graphene and n, m are integers. There are two high-symmetry directions termed ÔzigzagÕ and ÔarmchairÕ and are designated by (n; 0) and (n; n), respectively. SWCNTs are metals when (n m)/3 is an integer, otherwise they are semiconductors [6]. SWCNTs are promising materials as molecular containers with applications in the rechargeable lithium-batteries field because of their structure [8].
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00748-6
C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555
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Fig. 1. Schematics of the generation of a carbon nanotube by folding a section of a graphene sheet and the resulting nanotube is characterized by the chirality vector C. In the example shown here (dashed vector) C ¼ 3a1 þ 2a2 and the tube is labeled (3,2).
For instance, of the several forms of carbon, graphite is to date the best material to intercalate lithium and such systems are known as graphite intercalation compounds (GICs) [9,10]. Under optimal conditions lithium intercalates into graphite to form LiC6 , which leads to a modest specific capacity reduction potential of 372 mA h/g at the anode [11]. Obviously, the performance of these batteries would be considerably enhanced increasing the Li/C ratio of the host material. As previously mentioned, SWCNTs are formed when a graphene sheet is folded forming a cylinder. These cylinders usually aggregate into bundles, which consist of SWCNTs held together by van der Waals forces [12]. These bundles are expected to display a higher ability for intercalating lithium atoms and consequently higher energy storage capacity. In the ideal case, this gives an enhanced anode stoichiometry of LiC2 [13]. The superior battery performance of SWCNTs depends on the ability of lithium ions to enter and leave the nanotube interior at a reasonable rate. This rate can be improved if the lithium ion reach the interior through topological defects on the side walls and open-ends. As a matter of fact, in the experiments carried out by Gao et al. [14] the intercalation density was improved up to Li2:6 C6 after ball milling, suggesting that the ball milling process creates defects and breaks the nanotube, allowing the lithium ions to intercalate inside the nanotube [15].
In this Letter we use ab initio calculations to study the insertion of lithium ion through the side wall of SWCNTs. We have fully optimized three ÔarmchairÕ nanotubes with topological defects displaying 8, 9 and 10-membered rings. In comparison with previous theoretical investigations that have used a model with fewer atoms [16], we have fully optimized the nanotubes and obtained the corresponding geometries and wavefunctions. We have used the wavefunction to compute the molecular interaction potential (MIP) [17] and the MIP with polarization (MIPp) [18] of the nanotubes interacting with lithium cation. MIP and MIPp are valuable tools for exploring molecular reactivity and for rationalizing and predicting molecular interactions [19,20], including ion–p interactions [21–23]. MIPp is an improved generalization of the molecular electrostatic potential (MEP) [24], in which three terms contribute to the interaction energy: (1) an electrostatic term identical to the MEP, (2) a classical van der Waals dispersion–repulsion term [25] and (3) a polarization term derived from perturbational theory [26]. Therefore, it provides a natural partitioning of the interaction energy into intuitive components. It has been reported that the lithium ion motion in nanotubes is dominated by electrostatic effects [27]. Using MIP (sum of MEP and van de Waals terms) and MIPp (sum of MIP and polarization term) calculations we have analyzed the physical nature of the nanotube–lithium cation interaction,
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and determined if polarization is important by means of the computation of its contribution to the total interaction energy. Standard optimized van der Waals parameters [28] were used to describe the steric interactions of the classical particle (Liþ ).
energy in about 5 kcal/mol respecting the MP2 one (Fig. 2, right). Taking into consideration these results, the HF wavefunction was deemed suitable to carry out the MIP study involving the nanotubes described below. 3.2. Molecular interaction potential analysis
2. Computational methods Initially the geometries 2 of all SWCNTs included in this study were optimized at the restricted Hartree–Fock level using the PM3 semiempirical SCF-MO method [29] as implemented in the MO P A C -93 package [30] by means of the eigenvector following routine [31]. These structures have been used as starting points for the optimization at the HF/4-31G level of theory using the GA U S S I A N 98 program [32]. MIP calculations were computed using the MO P E T E computer program [33].
3. Results and discussion 3.1. Preliminary calculations In order to test that the Hartree–Fock (HF) wavefunction is useful to compute the MIP maps and consequently to analyze the molecular interaction potential of the nanotubes interacting with lithium cation we have carried out calculations on a model system. We have optimized a (4,4) ÔarmchairÕ nanotube with just one row of six-membered rings at the HF/4-31G and we have obtained the HF wavefunction. We have also performed a single point calculation at the MP2/4-31G//HF/4-31G level of theory with the purpose of obtaining the correlated (MP2) wavefunction. In Fig. 2 we represent the computed 2D-MIP energy maps of the model nanotube using the HF and the MP2 wavefunctions interacting with lithium cation. It can be observed that both maps are topologically very similar, the main difference is energetic. The HF map (Fig. 2, left) overestimates the interacting 2 Geometries are available upon request by contacting [email protected].
The HF/4-31G optimized geometries of nanotubes 1–3 are present in Fig. 3. They are (5,5) open-ended ÔarmchairÕ nanotubes with topological defects consisting in 8, 9 and 10-membered rings, respectively. First of all, we have computed bidimensional (2D) MIP energy maps of the nanotubes interacting with Liþ as a probe particle from the HF/ 4-31G wavefunctions. The 2D-MIP maps are shown in Fig. 4, the dashed isocontour lines correspond to negative interaction potential and solid isocontour lines to positive interaction potential. The 2D-MIP maps have been computed in a plane that bisects the nanotube as it is shown in Fig. 4. For nanotube 1, the computed 2D-MIP energy map shows negative (attractive) isopotential lines inside and outside the nanotube (see Fig. 4, left). Inside the nanotube, there is an extended region where the potential is )15 kcal/mol indicating that the lithium motion inside the nanotube is allowed. The lithium cations can enter through open-ends of the nanotube, however, the results obtained from MIP calculations indicate that the Liþ can not enter through the eight-membered topological defect. Moreover, the computed interaction potential outside the nanotube is about )10 kcal/mol, indicating that the interaction energy of Liþ – nanotube inside is more favorable than outside. However, in a realistic situation, when the lithium cation is placed outside, it is intercalated into the channels between nanotubes (bundle) interacting with several tubes simultaneously [13]. The behavior of nanotube 2 is similar to 1, negative interaction potential is observed inside and outside the nanotube and at the open-ends. The 2D-MIP energy map computed for 2 (see Fig. 4, middle) also indicates that the movement of the lithium ion through the nine-membered ring is not allowed. Finally, the 2D-MIP energy map of nanotube 3 has also been obtained (Fig. 4, right).
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Fig. 2. 2D-MIP (Liþ ) energy maps (HF on the left and MP2 on the right) obtained for the model ÔarmchairÕ nanotube (4,4). Negative isocontour dashed lines are shown every 5 kcal/mol and positive solid lines every 200 kcal/mol starting from 0 kcal/mol. The axes of the . cartesian coordinates are in A
Fig. 3. HF/4-31G optimized structures of nanotubes 1–3, displaying different topological defects.
Again, attractive interaction potential with lithium cation is observed at both interior and exterior of the nanotube, however, the lithium motion through the topological defect (10-membered hole) is not permitted. Nevertheless, in comparison with nanotubes 1 and 2, the diffusion barrier is lower. For compounds 1 and 2 no isopotential lines are observed in the 2D-MIP energy maps connecting the interior with the exterior of the nanotube, indicating that the barrier is higher than 100 kcal/
mol (the cutoff used to create the maps). In contrast, for nanotube 3 an isopotential line connecting the outside with the inside of the nanotube is observed at approximately 50 kcal/mol. 3.3. Molecular interaction potential with polarization analysis Cubero et al. [21] have demonstrated the importance of cation ! aromatic polarization effects
552 C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555 Fig. 4. 2D-MIP (Liþ ) energy maps obtained for 1–3. Negative isocontour lines are shown every 5 kcal/mol and positive lines every 20 kcal/mol starting from 10 kcal/mol. . The axes of the cartesian coordinates are in A
C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555
on cation–p interactions. Using theoretical calculations, they have demonstrated that polarization contribution is large in cation–aromatic interactions. Since MIP calculations do not take into account the polarization contribution to the total interaction energy, we have used the MIPp method which includes the polarization correction. As stated above, the polarization is accounted following a perturbation treatment. Computationally, due to the size of our systems, the calculation of 2D-MIPp energy maps requires an unaffordable increase in computer time. Therefore, instead of obtaining 2D-MIPp energy maps, we have computed the MIPp energies at several selected points. Even so, in order to test the validity of the MIPp method, we have obtained the 2D-MIPp energy map of the model nanotube (4,4). The obtained energy map is shown in Fig. 5, left. It can be observed that the topology is very similar to the computed 2D-MIP maps that are shown in Fig. 2. However, the potential interaction energy obtained for the MIPp map is considerably more negative than the obtained for the 2D-MIP maps, either HF or MP2. The MIPp minimum is located approximately at the geometrical center with an interacting energy of )55.6 kcal/mol which compares very satisfactorily with the computed interacting energy obtained from a full calculation of the model nanotube with a lithium cation at HF/4-
553
31G that is )58.9 kcal/mol. These results indicate that MIPp is a valuable and reliable tool for studying these systems. Calculation of the MIPp of the nanotubes 1–3 interacting with Liþ (see Table 1) was performed with the HF/4-31G geometries and wavefunctions. In the calculations, the Liþ ion was considered as a classical nonpolarizable particle. We have explored the electrostatic (Ee ), polarization (Ep ), van der Waals (Evw ), and total (Et ) interacting energies when a lithium cation is placed inside the Table 1 Contribution to the total interaction energy (kcal/mol) calculated with MIPp for compounds 1–3 interacting with Liþ at several points (see Fig. 5) Compound
Ee
Ep
Evw
Et
1, 1, 1, 1,
A B C D
)12.01 )6.91 )11.28 24.79
)30.54 )11.85 )24.96 )131.68
)4.23 )0.58 )3.12 375.36
)46.78 )19.34 )39.35 268.47
2, 2, 2, 2,
A B C D
)7.06 )5.98 )5.57 9.81
)42.07 )14.28 )28.68 )121.54
)4.34 )0.59 )3.34 121.42
)53.48 )20.85 )37.53 9.69
3, 3, 3, 3,
A B C D
)11.71 )6.87 )10.10 )4.06
)31.58 )13.66 )38.90 )93.53
)3.78 )0.59 )4.01 41.22
)47.07 )21.11 )53.01 )56.37
Fig. 5. On the left: 2D-MIPp (Liþ ) energy map obtained for the model ÔarmchairÕ nanotube (4,4). Negative isocontour dashed lines are shown every 10 kcal/mol and positive solid lines every 200 kcal/mol starting from 0 kcal/mol. The axes of the cartesian coordinates are . On the right: schematic representation of the four points where the MIPp has been computed. in A
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nanotube (point A, see Fig. 5, right), outside from the side the nanotube (point B, at 3.67 A wall), at the geometrical center of an open-end (point C) and at the center of the topological defect (point D). The results present in Table 1 point out the importance of the contribution of the polarization term to the total interaction energy. In all cases the polarization contribution is greater inside the nanotube than outside, indicating that the diffusion of the lithium cation inside the nanotube is favored by the polarization term. For nanotubes 1 and 2, the van der Waals term (Evw ) computed at the geometrical center of the topological defect prevents the diffusion of the lithium ion through eight- and nine-membered holes. The computed (MIPp) diffusion barrier of lithium through the enneagon is 9.69 kcal/mol (nanotube 2), similar to the obtained by Meunier et al. [11] (11.53 kcal/ mol) using ab initio density functional-based calculations with a real-space grid as a basis [34]. For nanotube 3, the computed MIPp value at point D is negative indicating that the lithium can move into the nanotube via the 10-membered hole. There is no energy barrier for the lithium ions to enter through the open-ends or topological defect consisting in at least decagon. Actually, there is a more negative potential of interaction inside showing that the nanotubes could act as attractors for lithium ions. Once inside their motion is not limited. For all nanotubes studied here, the interaction potential is always negative at points A, B and C, being more negative inside (A) than outside (B). Interestingly, the polarization term is much bigger at point D (topological defect) than at the other points, indicating that the motion of the lithium through the topological defect is favored by the cation-induced polarization energy. In summary, we have carried out theoretical investigation on the lithium diffusion in several SWCNTs which display different topological defects. Experimentally [14,15], a ball milling treatment of the nanotubes increases their capacity to storage lithium ions via inducing defects in the side walls and reducing the nanotube length. In this Letter we have studied which defects help the diffusion of the lithium by means of ab initio calculations and using the MIP and MIPp method-
ologies. We have found that lithium ions can enter nanotubes through topological defects consisting in a 10-membered ring and that the barrier for the diffusion through the nine-membered ring is just 9.69 kcal/mol. Finally, concerning the physical nature of the interaction between the lithium cation and nanotubes, the results in Table 1 point out the importance of the polarization contribution to the total interaction energy.
Acknowledgements Gratitude is expressed to the DGICYT and Conselleria dÕInnovaci o i Energia (Govern Balear) of Spain (projects BQU2002-04651 and PRDIB2002GC1-05, respectively) for financial support. We thank the Centre de Supercomputaci o de Catalunya (CESCA) for computational facilities. C.G. thanks the MECD for a predoctoral fellowship. We thank Prof. Marc Carbonell (UIB) for technical assistance. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
[10] [11] [12] [13] [14] [15] [16] [17] [18]
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C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555 [19] B. Hern andez, M. Orozco, F.J. Luque, J. Comput.-Aided Mol. Des. 11 (1997) 153. [20] D. Qui~ nonero et al., Chem. Phys. Lett. 326 (2000) 247. [21] E. Cubero, F.J. Luque, M. Orozco, Proc. Natl. Acad. Sci. USA 95 (1998) 5976. [22] D. Qui~ nonero et al., Angew. Chem. Int. Ed. 41 (2002) 3389. [23] D. Qui~ nonero et al., Chem. Phys. Lett. 359 (2002) 486. [24] E. Scrocco, J. Tomasi, J. Top. Curr. Chem. 42 (1973) 95. [25] M. Orozco, F.J. Luque, in: J.S. Murray, K. Sen (Eds.), Molecular Electrostatic Potential: Concepts and Applications, vol. 3, Elsevier, Amsterdam, 1996, p. 181. [26] M.M. Francl, J. Phys. Chem. 89 (1985) 428.
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Lithium diffusion in single-walled carbon nanotubes: a theoretical study Carolina Garau, Antonio Frontera *, David Qui~ nonero, Antoni Costa, Pau Ballester, Pere M. Deya 1 Departament de Quimica, Universitat de les Illes Balears, Crta. de Valldemossa km 7,5 07122 Palma de Mallorca, Spain Received 11 March 2003; in final form 11 April 2003
Abstract Theoretical investigations on single-walled carbon nanotubes (SWCNT) have been carried out using ab initio and molecular interaction potential with and without polarization (MIPp and MIP, respectively) methodologies to explore the possibilities of Liþ ion insertion through the side-wall of the nanotube. Li-nanotube systems can improve the capacity of lithium batteries by using both nanotube exteriors and interiors. Ab inito calculations of the fully optimized nanotubes were used to examine the topological defects depending on the ring size. The movement of the lithium cation through 8, 9 and 10-membered rings is studied and discussed. Ó 2003 Elsevier Science B.V. All rights reserved.
1. Introduction Carbon nanotubes [1,2] have a wide range of structural, mechanical and electronic properties [3]. Mainly, they represent a new class of materials for exploring nanoelectronics and molecular electronics [4,5]. A single-walled carbon nanotubes (SWCNT) can be either metallic or semiconducting material [6], depending on its diameter and chirality. This unique ability to be both metallic and semiconducting without doping has led to speculation that SWCNT might serve as a key building block for the construction and design of carbon-based electronics [4,5]. For instance, one *
1
Corresponding author. Fax: +34-971-173426. E-mail address: [email protected] (A. Frontera). Also corresponding author.
promising direction for the transistors of the future involves molecular electronics in which the active part of the device is composed of a single or a few molecules [7]. A SWCNT can be viewed as a seamless cylinder obtained by rolling-up a section of a two-dimensional graphene sheet (Fig. 1). It is uniquely characterized by the roll-up vector C ¼ na1 þ ma2 ¼ ðn; mÞ, where a1 and a2 are the primitive lattice vectors of the graphene and n, m are integers. There are two high-symmetry directions termed ÔzigzagÕ and ÔarmchairÕ and are designated by (n; 0) and (n; n), respectively. SWCNTs are metals when (n m)/3 is an integer, otherwise they are semiconductors [6]. SWCNTs are promising materials as molecular containers with applications in the rechargeable lithium-batteries field because of their structure [8].
0009-2614/03/$ - see front matter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0009-2614(03)00748-6
C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555
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Fig. 1. Schematics of the generation of a carbon nanotube by folding a section of a graphene sheet and the resulting nanotube is characterized by the chirality vector C. In the example shown here (dashed vector) C ¼ 3a1 þ 2a2 and the tube is labeled (3,2).
For instance, of the several forms of carbon, graphite is to date the best material to intercalate lithium and such systems are known as graphite intercalation compounds (GICs) [9,10]. Under optimal conditions lithium intercalates into graphite to form LiC6 , which leads to a modest specific capacity reduction potential of 372 mA h/g at the anode [11]. Obviously, the performance of these batteries would be considerably enhanced increasing the Li/C ratio of the host material. As previously mentioned, SWCNTs are formed when a graphene sheet is folded forming a cylinder. These cylinders usually aggregate into bundles, which consist of SWCNTs held together by van der Waals forces [12]. These bundles are expected to display a higher ability for intercalating lithium atoms and consequently higher energy storage capacity. In the ideal case, this gives an enhanced anode stoichiometry of LiC2 [13]. The superior battery performance of SWCNTs depends on the ability of lithium ions to enter and leave the nanotube interior at a reasonable rate. This rate can be improved if the lithium ion reach the interior through topological defects on the side walls and open-ends. As a matter of fact, in the experiments carried out by Gao et al. [14] the intercalation density was improved up to Li2:6 C6 after ball milling, suggesting that the ball milling process creates defects and breaks the nanotube, allowing the lithium ions to intercalate inside the nanotube [15].
In this Letter we use ab initio calculations to study the insertion of lithium ion through the side wall of SWCNTs. We have fully optimized three ÔarmchairÕ nanotubes with topological defects displaying 8, 9 and 10-membered rings. In comparison with previous theoretical investigations that have used a model with fewer atoms [16], we have fully optimized the nanotubes and obtained the corresponding geometries and wavefunctions. We have used the wavefunction to compute the molecular interaction potential (MIP) [17] and the MIP with polarization (MIPp) [18] of the nanotubes interacting with lithium cation. MIP and MIPp are valuable tools for exploring molecular reactivity and for rationalizing and predicting molecular interactions [19,20], including ion–p interactions [21–23]. MIPp is an improved generalization of the molecular electrostatic potential (MEP) [24], in which three terms contribute to the interaction energy: (1) an electrostatic term identical to the MEP, (2) a classical van der Waals dispersion–repulsion term [25] and (3) a polarization term derived from perturbational theory [26]. Therefore, it provides a natural partitioning of the interaction energy into intuitive components. It has been reported that the lithium ion motion in nanotubes is dominated by electrostatic effects [27]. Using MIP (sum of MEP and van de Waals terms) and MIPp (sum of MIP and polarization term) calculations we have analyzed the physical nature of the nanotube–lithium cation interaction,
550
C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555
and determined if polarization is important by means of the computation of its contribution to the total interaction energy. Standard optimized van der Waals parameters [28] were used to describe the steric interactions of the classical particle (Liþ ).
energy in about 5 kcal/mol respecting the MP2 one (Fig. 2, right). Taking into consideration these results, the HF wavefunction was deemed suitable to carry out the MIP study involving the nanotubes described below. 3.2. Molecular interaction potential analysis
2. Computational methods Initially the geometries 2 of all SWCNTs included in this study were optimized at the restricted Hartree–Fock level using the PM3 semiempirical SCF-MO method [29] as implemented in the MO P A C -93 package [30] by means of the eigenvector following routine [31]. These structures have been used as starting points for the optimization at the HF/4-31G level of theory using the GA U S S I A N 98 program [32]. MIP calculations were computed using the MO P E T E computer program [33].
3. Results and discussion 3.1. Preliminary calculations In order to test that the Hartree–Fock (HF) wavefunction is useful to compute the MIP maps and consequently to analyze the molecular interaction potential of the nanotubes interacting with lithium cation we have carried out calculations on a model system. We have optimized a (4,4) ÔarmchairÕ nanotube with just one row of six-membered rings at the HF/4-31G and we have obtained the HF wavefunction. We have also performed a single point calculation at the MP2/4-31G//HF/4-31G level of theory with the purpose of obtaining the correlated (MP2) wavefunction. In Fig. 2 we represent the computed 2D-MIP energy maps of the model nanotube using the HF and the MP2 wavefunctions interacting with lithium cation. It can be observed that both maps are topologically very similar, the main difference is energetic. The HF map (Fig. 2, left) overestimates the interacting 2 Geometries are available upon request by contacting [email protected].
The HF/4-31G optimized geometries of nanotubes 1–3 are present in Fig. 3. They are (5,5) open-ended ÔarmchairÕ nanotubes with topological defects consisting in 8, 9 and 10-membered rings, respectively. First of all, we have computed bidimensional (2D) MIP energy maps of the nanotubes interacting with Liþ as a probe particle from the HF/ 4-31G wavefunctions. The 2D-MIP maps are shown in Fig. 4, the dashed isocontour lines correspond to negative interaction potential and solid isocontour lines to positive interaction potential. The 2D-MIP maps have been computed in a plane that bisects the nanotube as it is shown in Fig. 4. For nanotube 1, the computed 2D-MIP energy map shows negative (attractive) isopotential lines inside and outside the nanotube (see Fig. 4, left). Inside the nanotube, there is an extended region where the potential is )15 kcal/mol indicating that the lithium motion inside the nanotube is allowed. The lithium cations can enter through open-ends of the nanotube, however, the results obtained from MIP calculations indicate that the Liþ can not enter through the eight-membered topological defect. Moreover, the computed interaction potential outside the nanotube is about )10 kcal/mol, indicating that the interaction energy of Liþ – nanotube inside is more favorable than outside. However, in a realistic situation, when the lithium cation is placed outside, it is intercalated into the channels between nanotubes (bundle) interacting with several tubes simultaneously [13]. The behavior of nanotube 2 is similar to 1, negative interaction potential is observed inside and outside the nanotube and at the open-ends. The 2D-MIP energy map computed for 2 (see Fig. 4, middle) also indicates that the movement of the lithium ion through the nine-membered ring is not allowed. Finally, the 2D-MIP energy map of nanotube 3 has also been obtained (Fig. 4, right).
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Fig. 2. 2D-MIP (Liþ ) energy maps (HF on the left and MP2 on the right) obtained for the model ÔarmchairÕ nanotube (4,4). Negative isocontour dashed lines are shown every 5 kcal/mol and positive solid lines every 200 kcal/mol starting from 0 kcal/mol. The axes of the . cartesian coordinates are in A
Fig. 3. HF/4-31G optimized structures of nanotubes 1–3, displaying different topological defects.
Again, attractive interaction potential with lithium cation is observed at both interior and exterior of the nanotube, however, the lithium motion through the topological defect (10-membered hole) is not permitted. Nevertheless, in comparison with nanotubes 1 and 2, the diffusion barrier is lower. For compounds 1 and 2 no isopotential lines are observed in the 2D-MIP energy maps connecting the interior with the exterior of the nanotube, indicating that the barrier is higher than 100 kcal/
mol (the cutoff used to create the maps). In contrast, for nanotube 3 an isopotential line connecting the outside with the inside of the nanotube is observed at approximately 50 kcal/mol. 3.3. Molecular interaction potential with polarization analysis Cubero et al. [21] have demonstrated the importance of cation ! aromatic polarization effects
552 C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555 Fig. 4. 2D-MIP (Liþ ) energy maps obtained for 1–3. Negative isocontour lines are shown every 5 kcal/mol and positive lines every 20 kcal/mol starting from 10 kcal/mol. . The axes of the cartesian coordinates are in A
C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555
on cation–p interactions. Using theoretical calculations, they have demonstrated that polarization contribution is large in cation–aromatic interactions. Since MIP calculations do not take into account the polarization contribution to the total interaction energy, we have used the MIPp method which includes the polarization correction. As stated above, the polarization is accounted following a perturbation treatment. Computationally, due to the size of our systems, the calculation of 2D-MIPp energy maps requires an unaffordable increase in computer time. Therefore, instead of obtaining 2D-MIPp energy maps, we have computed the MIPp energies at several selected points. Even so, in order to test the validity of the MIPp method, we have obtained the 2D-MIPp energy map of the model nanotube (4,4). The obtained energy map is shown in Fig. 5, left. It can be observed that the topology is very similar to the computed 2D-MIP maps that are shown in Fig. 2. However, the potential interaction energy obtained for the MIPp map is considerably more negative than the obtained for the 2D-MIP maps, either HF or MP2. The MIPp minimum is located approximately at the geometrical center with an interacting energy of )55.6 kcal/mol which compares very satisfactorily with the computed interacting energy obtained from a full calculation of the model nanotube with a lithium cation at HF/4-
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31G that is )58.9 kcal/mol. These results indicate that MIPp is a valuable and reliable tool for studying these systems. Calculation of the MIPp of the nanotubes 1–3 interacting with Liþ (see Table 1) was performed with the HF/4-31G geometries and wavefunctions. In the calculations, the Liþ ion was considered as a classical nonpolarizable particle. We have explored the electrostatic (Ee ), polarization (Ep ), van der Waals (Evw ), and total (Et ) interacting energies when a lithium cation is placed inside the Table 1 Contribution to the total interaction energy (kcal/mol) calculated with MIPp for compounds 1–3 interacting with Liþ at several points (see Fig. 5) Compound
Ee
Ep
Evw
Et
1, 1, 1, 1,
A B C D
)12.01 )6.91 )11.28 24.79
)30.54 )11.85 )24.96 )131.68
)4.23 )0.58 )3.12 375.36
)46.78 )19.34 )39.35 268.47
2, 2, 2, 2,
A B C D
)7.06 )5.98 )5.57 9.81
)42.07 )14.28 )28.68 )121.54
)4.34 )0.59 )3.34 121.42
)53.48 )20.85 )37.53 9.69
3, 3, 3, 3,
A B C D
)11.71 )6.87 )10.10 )4.06
)31.58 )13.66 )38.90 )93.53
)3.78 )0.59 )4.01 41.22
)47.07 )21.11 )53.01 )56.37
Fig. 5. On the left: 2D-MIPp (Liþ ) energy map obtained for the model ÔarmchairÕ nanotube (4,4). Negative isocontour dashed lines are shown every 10 kcal/mol and positive solid lines every 200 kcal/mol starting from 0 kcal/mol. The axes of the cartesian coordinates are . On the right: schematic representation of the four points where the MIPp has been computed. in A
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C. Garau et al. / Chemical Physics Letters 374 (2003) 548–555
nanotube (point A, see Fig. 5, right), outside from the side the nanotube (point B, at 3.67 A wall), at the geometrical center of an open-end (point C) and at the center of the topological defect (point D). The results present in Table 1 point out the importance of the contribution of the polarization term to the total interaction energy. In all cases the polarization contribution is greater inside the nanotube than outside, indicating that the diffusion of the lithium cation inside the nanotube is favored by the polarization term. For nanotubes 1 and 2, the van der Waals term (Evw ) computed at the geometrical center of the topological defect prevents the diffusion of the lithium ion through eight- and nine-membered holes. The computed (MIPp) diffusion barrier of lithium through the enneagon is 9.69 kcal/mol (nanotube 2), similar to the obtained by Meunier et al. [11] (11.53 kcal/ mol) using ab initio density functional-based calculations with a real-space grid as a basis [34]. For nanotube 3, the computed MIPp value at point D is negative indicating that the lithium can move into the nanotube via the 10-membered hole. There is no energy barrier for the lithium ions to enter through the open-ends or topological defect consisting in at least decagon. Actually, there is a more negative potential of interaction inside showing that the nanotubes could act as attractors for lithium ions. Once inside their motion is not limited. For all nanotubes studied here, the interaction potential is always negative at points A, B and C, being more negative inside (A) than outside (B). Interestingly, the polarization term is much bigger at point D (topological defect) than at the other points, indicating that the motion of the lithium through the topological defect is favored by the cation-induced polarization energy. In summary, we have carried out theoretical investigation on the lithium diffusion in several SWCNTs which display different topological defects. Experimentally [14,15], a ball milling treatment of the nanotubes increases their capacity to storage lithium ions via inducing defects in the side walls and reducing the nanotube length. In this Letter we have studied which defects help the diffusion of the lithium by means of ab initio calculations and using the MIP and MIPp method-
ologies. We have found that lithium ions can enter nanotubes through topological defects consisting in a 10-membered ring and that the barrier for the diffusion through the nine-membered ring is just 9.69 kcal/mol. Finally, concerning the physical nature of the interaction between the lithium cation and nanotubes, the results in Table 1 point out the importance of the polarization contribution to the total interaction energy.
Acknowledgements Gratitude is expressed to the DGICYT and Conselleria dÕInnovaci o i Energia (Govern Balear) of Spain (projects BQU2002-04651 and PRDIB2002GC1-05, respectively) for financial support. We thank the Centre de Supercomputaci o de Catalunya (CESCA) for computational facilities. C.G. thanks the MECD for a predoctoral fellowship. We thank Prof. Marc Carbonell (UIB) for technical assistance. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
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