Encapsulation of Carbon Chain Molecules in Single-Walled Carbon

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Encapsulation of Carbon Chain Molecules in Single-Walled Carbon Nanotubes Riichi Kuwahara,†,‡ Yohei Kudo,† Tsuguo Morisato,‡ and Kaoru Ohno*,†,§ †

Department of Physics, Yokohama National University, 79-5 Tokiwadai, Yokohama 240-8501, Japan Accelrys K. K., Kasumigaseki Tokyu Building 17F, 3-7-1 Kasumigaseki, Chiyoda-ku, Tokyo 100-0013, Japan § Institute for Materials Research, Tohoku University, 2-1-1 Katahira, Sendai 980-8577, Japan ‡

bS Supporting Information ABSTRACT: The vacuum space inside carbon nanotubes offers interesting possibilities for the inclusion, transportation, and functionalization of foreign molecules. Using first-principles density functional calculations, we show that linear carbonbased chain molecules, namely, polyynes (CmH2, m = 4, 6, 10) and the dehydrogenated forms C10H and C10, as well as hexane (C6H14), can be spontaneously encapsulated in open-ended single-walled carbon nanotubes (SWNTs) with edges that have dangling bonds or that are terminated with hydrogen atoms, as if they were drawn into a vacuum cleaner. The energy gains when C10H2, C10H, C10, C6H2, C4H2, and C6H14 are encapsulated inside a (10,0) zigzag-shaped SWNT are 1.48, 2.04, 2.18, 1.05, 0.55, and 1.48 eV, respectively. When these molecules come inside a much wider (10,10) armchair SWNT along the tube axis, they experience neither an energy gain nor an energy barrier. They experience an energy gain when they approach the tube walls inside. Three hexane molecules can be encapsulated parallel to each other (i.e., nested) inside a (10,10) SWNT, and their energy gain is 1.98 eV. Three hexane molecules can exhibit a rotary motion. One reason for the stability of carbon chain molecules inside SWNTs is the large area of weak wave function overlap. Another reason concerns molecular dependence, that is, the quadrupolequadrupole interaction in the case of the polyynes and electron charge transfer from the SWNT in the case of the dehydrogenated forms. The very flat potential surface inside an SWNT suggests that friction is quite low, and the space inside SWNTs serves as an ideal environment for the molecular transport of carbon chain molecules. The present theoretical results are certainly consistent with recent experimental results. Moreover, the encapsulation of C10 makes an SWNT a (purely carbon-made) p-type acceptor. Another interesting possibility associated with the present system is the direction-controlled transport of C10H inside an SWNT under an external field. Because C10H has an electric dipole moment, it is expected to move under a gradient electric field. Finally, we derive the entropies of linear chain molecules inside and outside an open-ended SWNT to discuss the stability of including linear chain molecules inside an SWNT at finite temperatures.

’ INTRODUCTION Carbon nanotubes (CNTs) can contain a variety of foreign molecules. Experimentally, it has been found that single-walled carbon nanotubes (SWNTs) can contain pure and endohedral fullerenes; the resulting structures are called “peapods”.14 SWNTs containing water,5,6 organic,710 and polyyne1114 molecules have also been reported. Although polyynes (CmH2), carbon linear chains terminated by hydrogen atoms, have high reactivities, they are stabilized up to 450 °C inside CNTs.11,12 It has been also found that CNTs can contain long linear carbon chains,15 tweezers-shaped alkyl chains,16 and alkyl-chain-functionalized fullerenes.17 Among these, it is particularly interesting to note that the mobility of chain molecules inside CNTs has been directly observed by means of transmission electron microscopy (TEM).16 Although there are well-known inclusion compounds (supramolecules) such as cyclodextrin that include long chain molecules,18 CNTs could serve as a complementary long linear cavity that includes such molecules. In addition to it being highly r 2011 American Chemical Society

desirable to obtain direct experimental evidence of the spontaneous encapsulation of carbon chain molecules inside CNTs, an independent theoretical confirmation would also be quite important to clarify the mechanism of the spontaneous encapsulation. From a theoretical viewpoint, classical molecular dynamics (MD) or Monte Carlo (MC) simulations, for example, have been used to study the encapsulation of C60 inside SWNTs19 and the stability of carbonwatercarbon composites made of doublewalled CNTs containing a considerable number of water molecules.20 Although these studies provide interesting information about the encapsulation of foreign molecules inside SWNTs, it remains difficult to understand the change in intermolecular forces during encapsulation or to obtain well-qualified energetics from classical MD or MC simulations alone. In contrast, firstReceived: September 28, 2010 Revised: February 18, 2011 Published: May 04, 2011 5147

dx.doi.org/10.1021/jp109308w | J. Phys. Chem. A 2011, 115, 5147–5156

The Journal of Physical Chemistry A

Figure 1. Potential contour map for the positions of the right hydrogen atom of a polyyne (C10H2), the axis of which is parallel to the axis of an open-ended (10,0) SWNT without hydrogen termination. The coordinates (x,y) are the same as in Figure 2. The energy zero corresponds to the polyyne being well-separated from the SWNT. Green, red, and blue areas indicate energies in the vicinity of, far above, and far below the energy zero, respectively.

principles calculations can provide more reliable information on the electronic structures and total energies involved. To date, numerous first-principles calculations have been performed to study the electronic structures of molecules encapsulated inside SWNTs, including C60 peapods,21,22 water,23 acetylene,24,25 organic or organometallic compounds,26 and others. However, most of these first-principles studies were intended to study only the stability and energetics of these endohedral systems. In particular, there has been very little detailed study of the potential curves around the open edges of the CNTs or inside and outside CNTs for such molecules. Moreover, linear carbon-based chain molecules have not yet been considered on the basis of first principles, probably because the size of the system to be addressed becomes quite large. In such circumstances, it becomes highly desirable to understand the mechanism of encapsulation and the possibilities of realizing the one-dimensional molecular transport and functionalization of linear chain molecules inside CNTs by exploring the electronic structures and potential maps of these systems on the basis of first principles. In this article, detailed first-principles density functional calculations are carried out for large systems composed of a linear carbon-based chain molecule and an openended SWNT in which both edges have dangling bonds or are terminated with hydrogen atoms. For a linear carbon-based chain molecule, we consider polyynes of different lengths, CmH2 (m = 4, 6, 10), the dehydrogenated forms C10 and C10H, and hexane C6H14. The aim of this study is three-fold: (1) to show the possibility of spontaneous encapsulation of each of these molecules inside SWNTs, (2) to clarify the mechanism of encapsulation for each molecule, and (3) to demonstrate the nature of the cavity inside SWNTs. We have calculated the total energies of these systems for various possible configurations to determine the potential fields for these molecules. We also determined that all of these molecules can be spontaneously encapsulated inside SWNTs and can move freely inside them, behaving as they would in an almost-ideal one-dimensional cavity. We have also clarified the electronic structure of these systems and the mechanism of spontaneous encapsulation. Finally, we discuss one-dimensional transport inside SWNTs and other distinct properties of the

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Figure 2. Potential energy (eV) for various angles of a polyyne (C10H2) near the entrance of an open-ended (10,0) SWNT with and without hydrogen termination. The coordinates (x,y,θ) are indicated in Figure 1. The distance between the polyyne and the left edge of the SWNT is x = 3.0 Å. The energy zero corresponds to the polyyne being wellseparated from the SWNT.

chain@SWNT systems. (Note that the @ symbol indicates that the chain molecules are encapsulated in the nanotubes.)

’ METHODOLOGY For our first-principles calculations, we used the DMol3 program,27,28 which employs a linear combination of the atomic orbital (LCAO) method. The local spin density approximation (LSDA) with the PerdewWang (PW) correlation functional29,30 was used in the density functional theory (DFT). This functional is considered more accurate than other functionals such as the VoskoWilkNusair (VWN) functional,31 and it well reproduces the Green’s function Monte Carlo results32 in both the high- and low-density limits of a uniform electron gas. Although we did not explicitly take into account the van der Waals interaction,33 this functional can effectively treat interatomic bonding around the equilibrium distances.34 The atomic orbitals were obtained numerically as values on an atomic-centered sphericalpolar mesh, rather than as analytical functions (e.g., Gaussian orbitals). The radial and angular parts of the atomic orbitals were obtained by solving the atomic KohnSham equation numerically and by spherical harmonics, respectively. The accuracy and validity of this method were demonstrated in our previous study on the interaction between an iron (Fe) atom and an open-ended SWNT.35,36 We used the double numerical basis set with polarization functions (DNP). Basically, when a basis set is localized on atoms, as in the DMol3 code, different configurations will have different basis sets, which introduces error in the calculations. This is called basis set superposition error (BSSE), and it can cause a fictitious binding of molecules. However, as was demonstrated in an earlier work,37 this problem does not exist for the present DNP calculations. The size of the DNP basis set is comparable to that of Gaussian 6-31G**. We selected the realspace cutoff radius, which represents the range of numerical integrations, to be 3.7 Å. This value is large enough not to impact the accuracy of the calculations. Although we started our calculations by introducing a smearing parameter for the Fermi distribution function to get rough convergence in electronic 5148

dx.doi.org/10.1021/jp109308w |J. Phys. Chem. A 2011, 115, 5147–5156


Figure 3. Potential energy curves for C6H2 (diamonds), C6H14 (downward-pointing triangles), C10H2 (circles), C10H (upward-pointing triangles), and C10 (squares) on the tube axis of an open-ended (10,0) SWNT without hydrogen termination. We set x = 0 when the rightmost atom of the chain molecule (CmHn) reaches the left edge of the SWNT and x > 0 when this atom is inside the SWNT. We inserted C10H from the carbon end. The energy zero corresponds to CmHn being well-separated from the SWNT (x = 10 Å).

states, we gradually decreased its value and ultimately set it to zero to avoid the use of any approximation of smearing in our calculations. We performed the geometry optimization of each molecule and SWNT independently, using the same program. For example, the radius of the (10,0) SWNT is R = 3.9 Å, and its length is 40.9 Å, whereas the radius of the (10,10) SWNT is R = 6.9 Å, and its length is 23.3 Å. Each of these SWNTs is composed of 400 carbon atoms. The calculated ground state of the zigzag (10,0) and (12,0) SWNTs is an open-shell singlet; that is, the left and right edges are spin-up and spin-down, regardless of the edge termination with hydrogen atoms.38 In contrast, the armchair (6,6) and (10,10) SWNTs have a closed-shell singlet ground state. The lengths of C6H14, C6H2, C10H2, C10H, and C10 are 8.1, 8.5, 13.6, 12.6, and 11.5 Å, respectively. Hexane, C6H14, has a zigzag structure with angles of 112.4° and a C—C bond length of 1.51 Å. For the other carbon chain molecules, the bonds between adjacent carbon atoms are alternately triple and single, and the lengths of the C—C, CC, and C—H bonds are 1.34, 1.22, and 1.07 Å, respectively.

’ RESULTS AND DISCUSSION A proposed contour map for the polyyne C10H2 is presented in Figure 1, in which the polyyne is parallel to the axis of the openended (10,0) SWNT without hydrogen termination. The rightmost position of the polyyne (hydrogen atom) corresponds to the coordinates (x, y); also see the Supporting Information. We verified that this map does not change much even when the dangling bonds at the edge of the SWNT are terminated by hydrogen atoms. Because there is no notable energy hump (potential barrier) or energy dip (trapping site) around the entrance to the SWNT, one can see that the polyyne can be spontaneously encapsulated inside the SWNT. This result is of a general nature, but to account for the fact that the polyyne can approach from different angles, we also considered two cases in which the polyyne is tilted with respect to the tube; see Figure 2. One case is when the rightmost atom of the polyyne is located on the tube axis (y = 0.0 Å in Figure 2), and the other is when the

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Figure 4. Potential energy curves for a polyyne molecule, C6H2, on the tube axis of an open-ended (10,0) SWNT with hydrogen termination (solid circles) and without hydrogen termination (open circles). We set x = 0 when the rightmost hydrogen atom of the polyyne reaches the left edge of the SWNT (positions of hydrogen and carbon atoms of the open-ended SWNTs with and without hydrogen termination, respectively) and x > 0 when this atom is inside the SWNT. The energy zero corresponds to the polyyne being well-separated from the SWNT (x = 10 Å).

rightmost atom of the polyyne is located near the shallow local minimum ( 0.12 eV), due to the dangling bonds around the tube edge (the bluish area at x = 3.0 Å, y = 4.0 Å in Figures 1 and 2). We assumed that x > 0 (x < 0) when the rightmost atom is inside (outside) the SWNT. In both cases, the potential energy maintains its value in the range of tilt angles between 0° and 45° for an open-ended SWNT without hydrogen termination, but it increases by about 0.1 eV at around 45° for a hydrogen-terminated SWNT. This indicates that hydrogen termination at the edge of an SWNT yields an energy barrier around the 45° tilt angle, which consequently enhances the possibility of encapsulation at a small tilt angle. Under typical experimental conditions, the open edge of the SWNT would be at least partially terminated by hydrogen atoms, and in such cases, encapsulation could take place more readily. Figure 3 shows the calculated potential curves for C6Hn (n = 2, 14) and C10Hn (n = 2, 1, 0) located on the tube axis of the openended (10,0) SWNT without hydrogen termination. For the sixcarbon polyyne, C6H2, we verified that the potential curve became smoother when we introduced hydrogen termination at the edge of the SWNT; see Figure 4. In these figures, the parameter x is defined as the distance between the rightmost atom of the chain molecule (either a carbon or a hydrogen) and the left edge of the SWNT along the tube axis. We assumed that the rightmost atom of C10H is carbon. From these figures, it becomes clear that all of these molecules can be spontaneously encapsulated as if they were drawn into a vacuum cleaner. When C6H14 or CmH2 (m = 6, 10) enter an open-ended SWNT without hydrogen termination (Figure 3), the interaction potential is almost 0 eV for x < 0; it slightly decreases around x = 0 and then linearly decreases for x > 0. The energy decrease occurs somewhat more rapidly for C10H and C10 than for CmH2 (m = 6, 10), because there is a dangling bond at the carbon end of the linear chain molecules. The potential energy of hexane (C6H14) exhibits an intermediate behavior because the bond between the rightmost hydrogen and carbon atoms is inclined with respect to the tube axis and its length along the tube axis is 5149



Figure 5. Difference in the electron charge density (ECD) of CmHn@(10,0)SWNT compared to the two independent ECDs of isolated CmHn and the isolated (10,0) SWNT: (a) C10H2@SWNT, (b) C10H@SWNT, (c) C10@SWNT, and (d) C6H14@SWNT. Blue and yellow indicate the areas where the ECD increases and decreases, respectively, at inclusion.

shorter than that of C10H2. The potential energy starts to decrease almost when the rightmost carbon atom comes into the entrance of the SWNT. For open-ended SWNTs without hydrogen termination, when the left edge of the chain molecules comes close to the nanotube entrance, the potential energy reaches a minimum. Then, the potential energy increases very slightly, and when the chain molecules are fully encapsulated inside the SWNT, the potential energy saturates to a constant value. In contrast, for hydrogen-terminated SWNTs, the energy decreases monotonically to a constant value. The energy gains when C6H14, C6H2, C10H2, C10H, and C10 are encapsulated inside the (10,0) SWNT are 1.48, 1.05, 1.48, 2.04, and 2.18 eV, respectively. These values should be compared with the experimental and theoretical binding energy of the stable face-centered cubic (fcc) C60 crystal (∼1.7 eV/C60).3941 This is far larger than the values expected for pure van der Waals interaction. Generally, the LSDA functional cannot estimate the van der Waals interaction, although the general trend should be correct considering the error cancellation. However, the main interaction between C60 molecules comes from the weak wave function overlap and multipole interactions depending on the intermolecular orientations, which can be computed within the LSDA. According to our LSDA calculations, the binding energy was estimated to be 1.6 eV/C60. Figure 5 shows the difference in the electron charge density (ECD), when the ECD of CmHn inside the (10,0) SWNT is subtracted from the sum of the two independent ECDs of the isolated CmHn and isolated (10,0) SWNT. Blue and yellow indicate areas where the ECD increases and decreases, respectively, at inclusion. Clearly, quadrupole charge redistribution occurs in the polyyne (C10H2) and in the SWNT (Figure 5a).

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Therefore, the attractive interaction between the polyyne and the SWNT is at least partly due to this quadrupolequadrupole interaction. In the polyyne, 0.07 electron moves toward the center of the molecule at encapsulation, whereas in the SWNT, 0.18 electron is moved aside. On the other hand, electron charge transfer takes place from the SWNT to C10H (Figure 5b) and to C10 (Figure 5c); the charge transfer involves 0.7 and 0.9 electron, respectively. This finding is consistent with the shift in the empty minority-spin level of an isolated C10 down to the Fermi level when this C10 enters the SWNT (see the discussion associated with Figure 7 below). This electron charge transfer from the SWNT to C10 suggests that the resulting C10@SWNT behaves as a p-type acceptor. For hexane (C6H14), in Figure 5d, blue (electron-rich) areas connect the hexane molecule and the tube wall, stemming from the wave function overlap between them. As a result, this charge redistribution induces multipole charge distribution in the SWNT, although the main attractive interaction between hexane and the SWNT is due not to the resulting multipole but to the wave function overlap. We will discuss this again later. The magnitudes of the weak wave function overlap and multipole interactions are strongly dependent on the molecule and its relative geometry with respect to the SWNT, because the system tries to decrease the total energy by utilizing these interactions as effectively as possible. We estimated the Coulombic interaction energies between C10Hn and the SWNT by separating the ECD into C10Hn and SWNT sides onto separate halves of the tube radius, calculating the ECD differences between the two sides before and after inclusion, and using a fast Fourier transformation (FFT). We divided the three-dimensional space of 46.8 14.0 13.8 Å3 into a grid of 1536 768 768 and estimated the Coulombic interaction energies in Fourier space according to the equation U ¼

4π Ω

∑G

FC10 Hn ðGÞ FSWNT ðGÞ G2

ð1Þ

where Ω denotes the volume of the grid space. The estimated energies due to the electron charge transfer from the SWNT to C10H and C10 were 0.80 and 0.93 eV, respectively. For the quadrupolequadrupole interaction energy between the polyyne and the SWNT, we used the total charge density distribution on the polyyne side, because the polyyne has an inherent quadrupole. The estimated energy was 0.37 eV. The remainder of the energy gain is almost independent of n for n = 2, 1, and 0 and is about 1.15 eV, which is due to the weak overlap of the wave functions. The electron affinity of the isolated C10 molecule is 4.8 eV,42 and the ionization potential of the isolated (10,0) SWNT is 6.2 eV;43 a simple calculation states that the charge transfer of 0.9 electron from an SWNT to C10 requires (6.2 4.8) 0.9 = 1.26 eV. Therefore, even if a 0.93 eV energy gain is caused by the resulting Coulomb attraction, this charge transfer is energetically unfavorable unless there is a change in the electronic states on both sides. We therefore conclude that the wave function overlap and the change in the electronic states are essential causative factors in this charge transfer. Figure 6 shows the wave function corresponding to the doubly degenerate HOMO (highest occupied molecular orbital) level of the polyyne molecule (C10H2) located on the tube axis of the open-ended (10,0) SWNT without hydrogen termination. This state corresponds to the one with maximum overlap with the original HOMO of the molecule. When the polyyne is well outside the SWNT (x , 0), there is no wave function overlap 5150



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Figure 6. Wave function (KohnSham orbital) of a polyyne molecule (C10H2), which corresponds to the HOMO level of the polyyne. Wave function overlap appears as the polyyne is encapsulated in the SWNT, when the polyyne is located at several distances, x, on the tube axis of the SWNT. The LDA eigenvalue and the level relative to the HOMO level of the whole system are shown for each geometry. The Fermi energy is about 5.20 eV.

Figure 7. Wave functions of (left) C10@SWNT and (right) isolated C10 at two doubly degenerate levels near the Fermi level, where C10 has finite amplitude. The wave functions resemble each other for all four levels, including two degenerate spin-up levels and two degenerate spindown levels. The LSDA eigenvalues and the spin states are also shown. The LUMO level shifts below the Fermi level and becomes the HOMO level when C10 is included inside the SWNT, and therefore, an electron charge transfer from the SWNT to C10 takes place.

between the polyyne and the SWNT. However, when the right edge of polyyne comes close to the entrance of the SWNT at x = 0, a weak wave function overlap appears between them. Then, as the polyyne enters the SWNT, the area of wave function overlap increases. This increase is considered to be one of the causes of the energy gain when the polyyne is encapsulated inside an SWNT.

This dynamic is not peculiar to this system. For example, in the interaction between graphite layers, wave function overlap is considered the main cause of interlayer cohesion.41,44,45 When the left edge of the polyyne comes close to the entrance of the SWNT at x = 14 Å, the wave function overlap between the left hydrogen atom of the polyyne and the left dangling bonds of the SWNT becomes most prominent. Because the wave function overlap is at a maximum at this point, the total energy reaches a minimum here. After passing through this point, the total energy increases only very slightly, and there remains a wave function overlap between the polyyne and the inner π orbitals of the SWNT. The absolute value of the wave function (i.e., the bonding orbital) in the intermolecular region between the polyyne and the SWNT is about 1.2 102 Å1.5. This is due to the concentration of the inner centripetal π orbitals of the SWNT and the overlap with the π orbitals of the polyyne. This value is small but not negligible, because the overlapping area around the polyyne is definitely large. A similar wave function overlap also occurs for C10 and C10H, although we do not show as many figures for them in order to avoid redundancy. Instead, we show only the case in which C10 is located in the middle of the SWNT (at x = 26.2 Å), in the left half of Figure 7. Here, a new type of wave function overlap that was not seen in the case of C10H2 appears at the lower levels. This overlap occurs between the dangling bonds at both ends of C10 and the inner π orbitals of the SWNT. These wave functions correspond to those of the isolated C10 molecule shown in the right half of Figure 7. Here, we notice that, whereas the upper doubly degenerate level is half-occupied in an isolated C10 molecule (on the right) (C10 is therefore spin-polarized), threefourths of the corresponding level is occupied when it is inside the SWNT (on the left). That is, the up-spin LUMO level of C10 in the diagram on the right comes down below the Fermi level and becomes the HOMO level in the diagram on the left. It is for this reason that electron transfer occurs from the SWNT to C10. This 5151



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Figure 8. Potential energy curves for C10Hn (n = 2, 0) inside a (10,0) SWNT and C6Hn (n = 2, 14) inside a (10,10) SWNT, when the molecules are parallel to the tube axis. The diameters of the (10,0) and (10,10) SWNTs are 7.8 and 13.8 Å, respectively. The energy zero is set equal to the total energy when the molecules are sufficiently separated from the SWNTs. For C10Hn (n = 2, 0) inside a (10,0) SWNT, the potential is almost flat up to 0.8 Å, whereas for C6Hn (n = 2, 14) inside a (10,10) SWNT, there is no interaction between the molecules and the SWNT up to 1.5 Å.

Figure 9. Potential energy curves for C10Hn (n = 2, 0) outside a (10,0) SWNT and C6Hn (n = 2, 14) outside a (10,10) SWNT, when they are parallel to the tube. The energy zero is set equal to the total energy when the molecules are sufficiently separated from the SWNTs. The potential energies for C10H2, C10, and C6H2 reach minima at about 3.2 Å, whereas the potential energy for C6H14 reaches a minimum at 3.6 Å because of its fully hydrogenated structure. The energy gain of CmHn outside the SWNT is less than one-half the energy gain inside the SWNT.

also explains why the energy gain of C10 (2.2 eV) is larger than that of C10H2 (1.5 eV). Figure 8 shows the potential energies of C10Hn (n = 2, 0) inside a (10,0) SWNT and C6Hn (n = 2, 14) inside a (10,10) SWNT as functions of the off-center distance y (Å) when the molecules are parallel to the tube axis. The zero of the potential energy is the energy when the molecules are well outside the armchair SWNTs. Although the HOMOLUMO gap of armchair SWNTs is known to show an oscillatory behavior according to tube length46,47 (our HOMOLUMO gap energies were 0.60, 0.42, 0.07, 0.51, and 0.38 eV, respectively, for C320H40, C340H40, C360H40, C380H40, and C400H40), we found that the energy gain of C6Hn (n = 2, 14) at y = 3.5 Å inside the (10,10) SWNT with or without hydrogen termination, CmH40 or Cm (m = 320400), does not depend on the tube length and has a constant value within the error of 0.01 eV. For C10Hn (n = 2, 0) inside the (10,0) SWNT, there is a slightly greater energy gain of about 0.07 eV at y = 0.7 Å than at the center y = 0. However, this is a very subtle change. The potential energy shows almost no change in the interval 0 e y e 0.8 Å, and it rapidly increases for y > 0.8 Å. This implies that the space inside the SWNT behaves as a free space of a cylinder of radius 0.8 Å for the polyyne. For the C6Hn molecules (n = 2, 14) inside the (10,10) SWNT, there is no energy gain when they are located on the tube axis, but an energy gain appears when C6H2 (C6H14) is about 3.5 Å (3.0 Å) off the tube axis. On the other hand, when the carbon chain molecules are outside the SWNT surface, the potential energy reaches minimum at a distance of 3.2 Å (3.6 Å for C6H14) from the SWNT surface as shown in Figure 9. The energy gains at this point (measured from the energy at an infinite distance) are at least 0.3 eV for C6H14 and at most 0.7 eV for C10, which are much smaller values than the energy gains when these molecules are included inside the SWNT. The potential energy increases rapidly as the molecules move toward the SWNT surface, as expected. This result clearly demonstrates that chain molecules are energetically more favorable when they are contained inside an SWNT rather than

adsorbed on the SWNT surface. Thus, we conclude that they are spontaneously included inside the SWNT. The results for the energy gains of linear chain molecules at different positions inside the SWNTs are summarized in Table 1. The energy gains of C6H2, C4H2, and C2H2 at the center of the (10,0) SWNT are also listed in this table. The energy gain, ΔE in units of electronvolts, of CmH2 (m = 4, 6, 10) can be very well fitted with the formula ΔE = 0.405 þ 0.107m (note that this excludes acetylene, i.e., m = 2, C2H2). That is, the interaction between the polyyne and the (10,0) SWNT is 0.107 eV per polyyne carbon atom, with an additional constant of 0.405 eV. In this table, the energy gains of C6H2 are also reported for the center of the (6,6) armchair and the (12,0) zigzag SWNTs, as well as the (10,0) zigzag SWNT. Its energy gain, ΔE, decreases as the radius R of the SWNT increases. On the other hand, when C6H2 approaches the tube surface inside the (10,10) SWNT, the energy gain increases and becomes a maximum at an off-center distance of y = 3.5 Å. In the case of C6H14 inside the (10,10) SWNT, there are two possible optimal configurations, as shown in Figure 10: (a) lying and (b) standing. Table 1 indicates that configuration a (lying) is slightly more stable than configuration b (standing). We also investigated the case in which the molecular axis of the chain molecule is tilted with respect to the tube axis deep inside (in the middle of) the SWNT. Figure 11 shows the potential energies of C10Hn (n = 2, 1, 0) and C6H14 inside a (10,0) SWNT and of C6H14 inside a (10,10) SWNT as a function of the parameter θ, the angle between the two molecular axes. In this figure, one can see that the total energy does not change in the region |θ| e |θmax| with θmax ≈ 10° in the case of the (10,0) SWNT. This θmax value represents the maximum allowed tilt angle of the linear chain molecules with respect to the nanotube axis without a loss in energy. There is no θmax value for hexane (C6H14) inside the (10,10) SWNT. This result suggests that, when more than two chain molecules are included inside a (10,10) SWNT, their alignment could become inclined or nested. As an example, we calculated the results for a (10,10) SWNT including three hexane molecules simultaneously. We found that the most 5152



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Table 1. Calculated Energy Gains of CmHn Fully Encapsulated Inside a (k,l) SWNT radius, CNT

R (Å)

(10,0)

3.9

molecule

length of

off-center

energy

molecule

distance,a

gain,b

(Å)

y (Å)

ΔE (eV)

C10H2

13.6

0.0

1.48

C10H

12.6

0.0

2.04

C10

11.5

0.0

2.18

C6H2

8.5

0.0

1.05

C4H2

6.1

0.0

0.83

C2H2

3.5

0.0

0.55

C6H14

8.1

0.0

1.48

(6,6) (12,0)

4.1 4.7

C6H2 C6H2

8.5 8.5

0.0 0.0

0.74 0.26

(10,10)

6.9

C6H2

8.5

0.0

0.00

3.5

0.49

C6H14

8.1

0.0

0.00

3.1 (Figure 10a)c

0.59

3.0 (Figure 10b)c

0.48

Figure 11. Change in the potential energy when C10Hn (n = 2, 1, 0) or hexane (C6H14) is tilted inside an SWNT with angle θ. The tilt angle inside the (10,0) SWNT is plotted in the lower scale in the range 030°. The tilt angle of C6H14 inside the (10,10) SWNT is plotted as crosses in the upper scale in the range 090°.

a

Indicates the most stable position inside the (10,10) SWNT. b When CmHn is on the tube axis, the greater the diameter of the SWNT is, the smaller the energy gain is. When C6H2 and C6H14 are on the axis of a (10,10) SWNT, the energy gain is almost zero. c Note that, in the case of the (10,10) SWNT and C6H14, there are two optimal configurations, as shown in Figure 10. The most stable configuration is that of Figure 10a.

Figure 10. Two stable configurations of C6H14 inside a (10,10) SWNT: (a) lying and (b) standing. As shown in Table 1, configuration a is slightly more stable than configuration b.

stable configuration is such that the three hexane molecules are nested in a parallel orientation, with each molecule having the configuration shown in Figure 10a. The molecular geometry inside the (10,10) SWNT is depicted in Figure 12. In this configuration, the energy gain for the encapsulation of three hexane molecules is 1.98 eV, which is more than 3 times the encapsulation energy gain for a single hexane molecule (i.e., 3 0.59 = 1.77 eV) because of the additional intermolecular interactions among the hexane molecules. Therefore, this parallel configuration is more stable than the serial configuration in which hexane molecules are aligned inside the (10,10) SWNT. The ECD difference between this system and the sum of the three isolated hexane molecules and the SWNT was calculated, and its four different views are depicted in Figure 13. As in Figure 5d for hexane@(10,0)SWNT, electron-rich blue areas are widespread between the three hexane molecules and the SWNT wall. Moreover, views from different angles clearly show that the electron-poor yellow areas are spread over the skeleton parts of the hexane molecules and the π orbitals of the SWNT. This means that an electron moves from the

Figure 12. Image of a (10,10) SWNT encapsulating three hexane molecules in parallel orientation.

intramolecular yellow region to the intermolecular blue region. We thus conclude that the attractive interaction between hexane and an SWNT comes from a weak wave function overlap between the side hydrogen atoms of hexane and the carbon atoms of the SWNT in the blue region. In this figure, one can clearly see that multipole electron charge redistribution appears on both the hexane and nanotube sides. In particular, the SWNT exhibits a (blue and yellow) striped pattern parallel to the tube axis because of the parallel encapsulation of three hexane molecules. Because this multipole charge redistribution must be quite sensitive to the electric field, one might expect, for example, that the inner three hexane molecules would exhibit a rotary motion under circularly polarized light. We performed geometry optimization of C10Hn@SWNT (n = 0, 2) for C10Hn located at x = 27.2 Å and y = 0.0 Å [i.e., in the middle of the (10,0) SWNT], in order to check whether any change in the structure occurred after structural relaxation. No significant difference was observed between the structures before and after 5153



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Figure 13. Electron charge density (ECD) difference between the system of three hexane molecules encapsulated in parallel orientation in a (10,10) SWNT and the isolated molecules. The ECD difference is shown from four viewpoints: (i) distant front view from the tube axis (upper left), (ii) distant side view from the radial axis (upper right), (iii) close front view around C6H14 and the inner wall of the SWNT (lower left), and (iv) close side view around one hexane molecule (lower right). Blue and yellow indicate the areas where the ECD increases and decreases, respectively, at inclusion.

the geometry optimization. The maximum displacement occurred in the end atoms of C10H2 and C10 toward the SWNT surface, with displacements of 0.06 and 0.14 Å for C10H2 and C10, respectively. When we performed the geometry optimization starting from an initial geometry in which C10Hn (n = 0, 2) was located slightly off-center (y = 0.1 Å) from the SWNT axis, both molecules moved toward an off-center position around y = 0.7 Å, but the structures of the molecules and the SWNT did not change much. The maximum displacement on the SWNT side occurred in the open-edge carbon atoms, with displacements of 0.06 and 0.05 Å for C10H2@SWNT and C10@SWNT, respectively. The average displacement was 0.01 Å. In particular, the structure of the polyyne molecule was exactly the same both inside and outside the SWNT. This fact, together with the flatness of the potential curve inside the SWNT, suggests that the space inside the SWNT might present an ideal environment for the transport of linear chain molecules. Finally, we briefly comment on the entropy of linear chain molecules inside and outside an open-ended SWNT to address the stability of the chain@SWNT system at finite temperatures. Because the total volume of the free space outside the SWNT (Vout) is much larger than the total volume of free space inside the SWNT (Vin), chain molecules outside the SWNT have a larger entropy than those inside. One can calculate the rotational part of the partition function of a two-atom molecule using the Euler angle φi = (θ,φ,ψ)48 as Zrot ¼

2πkB TI φmax ð1 cos θmax Þ h2

ð2Þ

where I is the moment of inertia. Then, using φmax = θmax inside the SWNT and φmax = θmax = π outside the SWNT, the entropy

difference per chain molecule can be obtained as 2πVout ΔS ¼ Sout Sin ¼ kB ln ð3Þ πr 2 LSWNT θmax ð1 cos θmax Þ Here, LSWNT (m) is the total length of the SWNT per chain molecule; r (m) is the radius of the region inside the SWNT in which the potential energy for a chain molecule is at a maximum, which is almost constant around the tube axis; and Vout (m3) is the total volume outside the SWNT per chain molecule (1/Vout is the chain density). For example, for polyyne C10H2 and a (10,0) SWNT, the maximum tilt angle, θmax, is about 10°, and the radius of the free region inside the SWNT is r = 8.0 1011 m. Then, the free energy difference per polyyne molecule at temperature T is estimated as ! 1:2 1023 Vout ΔF ¼ Fout Fin ¼ ΔE kB T ln ð4Þ LSWNT The chain molecules are stable inside the SWNT for ΔF > 0 but unstable otherwise. In Nishide et al.’s Raman experiments, the polyyne peak disappeared above 450 °C, which corresponds to this transition. On the other hand, using kBT = 0.026 eV at room temperature and ΔE = 1.5 eV for C10H2 inside a (10,0) SWNT, we obtain Vout ΔF 0:12 0:026 ln ðeVÞ LSWNT Vout 0:026 ln ðeVÞ ð5Þ 102 LSWNT where ΔE denotes the potential energy gain after encapsulation. In this equation, the value of ΔE is combined with some number 5154


The Journal of Physical Chemistry A resulting from the decomposition of the logarithmic term to give 0.12 eV. Then, the criterion for which inclusion occurs is given by ΔF > 0. This corresponds to the condition Vout (m3) < 102LSWNT (m) for C10H2 inside a (10,0) SWNT. Because Vout is not the total volume but the volume per chain molecule, if the chain density outside the SWNT is rather high, this condition will be satisfied. Then, encapsulation of the polyyne molecules inside an SWNT will occur spontaneously at room temperature.

’ CONCLUSIONS In this work, we performed first-principles calculations of linear chain molecules inside and outside open-ended SWNTs and found that polyynes and their dehydrogenated forms, as well as hexane, can be spontaneously encapsulated inside an SWNT. When the polyyne molecules approach the dangling bonds of an SWNT, they might interact weakly with the dangling bonds. This result is consistent with experimental findings.11,15,16 One reason for the encapsulation is the large area of weak wave function overlap between the CmHn molecules and the SWNT. The other reason is not the same for these molecules, but concerns molecular dependence: the quadrupolequadrupole interaction in the case of CmH2 and electron charge transfer from the SWNT in the cases of CmH and Cm. Alkyl chains such as hexane (C6H14) can be encapsulated in a triangle-shaped parallel configuration of three molecules inside a (10,10) SWNT. The very flat potential surface inside an SWNT suggests that the friction is quite low, just as on a graphite surface,49,50 and that the space inside an SWNT provides an ideal environment for the molecular transport of linear chain molecules. Moreover, the encapsulation of Cm makes an SWNT into a (purely carbon made) p-type acceptor, similar to the result observed experimentally in tetracyanoquinodimethane (TCNQ).7 Another interesting possibility associated with the present system is the linear transport of CmH molecules inside an SWNT under an external field. Because CmH molecules have an electric dipole moment, one would expect that they would move in response to a gradient electric field. The entropy discussion provided in the last section of this article might help in understanding the possibility of realizing chain@SWNT systems at finite temperatures. ’ ASSOCIATED CONTENT

bS

Supporting Information. Original data used to draw the potential contour map (Figure 1) of polyyne around an SWNT. This material is available free of charge via the Internet at http:// pubs.acs.org.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ ACKNOWLEDGMENT The authors thank Dr. Tomonari Wakabayashi, Dr. Soh Ishii, and Dr. Hannes Raebiger for helpful discussions. This work was partly supported by Grants-in-Aid for Scientific Research B (Grants 17310067 and 21340115) from the Japan Society for the Promotion of Science.

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Encapsulation of Carbon Chain Molecules in Single-Walled Carbon

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