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J. Phys. Chem. C 2008, 112, 15260–15266
Classical Trajectory Studies of the D + H2 f HD + H Reaction Confined in Carbon Nanotubes: Parallel Trajectories Tun Lu† and Evelyn M. Goldfield* Department of Chemistry, Wayne State UniVersity, Detroit, Michigan 48202
Stephen K. Gray Chemical Sciences and Engineering DiVision, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: May 20, 2008; ReVised Manuscript ReceiVed: July 28, 2008
We use full-dimensional classical trajectories to study how reaction probabilities for the D + H2 f DH + H reaction are altered when the system is confined to move within various-sized carbon nanotubes (CNTs). This study focuses on D atoms initially moving parallel to the long axis of the tube. We compare our results with standard gas-phase reaction probabilities. Enhanced reaction probabilities are found for the smaller diameter CNTs, and slight quenching is found for the largest diameter CNT studied. These results are also consistent with those of a reduced-dimensional, quantum study. The origins of the confinement effects are discussed in terms of how the CNT modifies the H2 reactant state and of the modified forces experienced by the incoming D atom. I. Introduction Although the effect of confinement in nanoscale media on chemical reactivity has generated considerable recent interest,1-4 there has been very little theoretical research on the dynamics of such systems. The idea that carbon nanotubes (CNTs), for example, might be employed as “nano test tubes”1 is very intriguing. Dynamical studies that focus on very simple chemical reactions are required to explore the implications of this idea and to develop a theoretical framework for understanding the effects on confinement on chemical reactivity. In a previous study,5 which we will refer to as Paper I, we explored the quantum dynamics of the D + H2 f HD + H reaction confined to occur within CNTs of various sizes and structures. The H + H2 reaction (and its isotopic variants), one of the simplest of all chemical reactions, has played a fundamental role in chemical reaction dynamics both in terms of establishing concepts and in terms of detailed comparisons between theory and experiment. In general, the H + H2 system remains the subject of intense study.6-13 We reasoned that since the reaction has a collinear transition state one might expect that confinement within a cylindrical environment will enhance reactivity. Indeed, we found that confinement within small nanotubes of diameter e7.2 Å led to enhancement of reactivity at all collision energies compared to the isolated system. We also found a lowering of the reaction threshold in all nanotubes we studied, which we believe is a quantum effect. The quantum studies in Paper I contained a number of simplifications. We held the nanotube carbons rigid and used a reduced dimensionality model to describe the remaining degrees of freedom. We also ignored the short-range interactions between the H/D atoms and the nanotube carbon atoms, thereby using only the long-range Lennard-Jones potential terms. One might argue that the confinement effects we saw could be artificial effects due to the simplifications in our model. Thus,
we turn to classical dynamics where it is possible to eliminate the restrictions imposed in our quantum model. The classical trajectory method is computationally much cheaper but can still give accurate results if quantum effects are small. For example, classical and quantum reaction probabilities for the D + H2 reaction in the gas phase agree quite well.12 Our classical studies treat the full dimensionality of the confined system. The carbon atoms of the nanotube are permitted to move, and we employ a potential that treats both long-range and short-range interactions of the confined system with the nanotube carbons. Furthermore, classical trajectory studies allow one to explore in detail different models for choosing initial conditions in the hopes that one can tease apart the various contributions to the enhancement of reactivity. This study is the first of two studies on the classical dynamics of the D + Η2 f DΗ + H reaction confined in a carbon nanotube, which we refer to as D + H2@CNT. We consider three nanotubes characterized by differing chiral vectors (n, m)14 and associated diameters, d: (8, 0) with d ) 6.26 Å, (2, 8) with d ) 7.2 Å, and (6, 6) with d ) 8.14 Å. The model used in this work is such that the initial velocity of the incoming D atom is parallel to the long axis of the nanotube, and the model is more readily compared to the results of our quantum study than the more general models used in the second study. In this first study, we focus on how the preferred orientation of the reactants and of the interacting system with respect to the confining nanotube affects the reaction probability. Collisions with the walls of the CNT play a minor role. Such collisions, however, may contribute to enhanced reactivity. In the second study, which will be detailed in a subsequent publication, more general models are considered. In these models, the D atom does not move parallel to the CNT axis, and collisions with the nanotube wall play a much more important role. II. Methods
* Corresponding author. E-mail:
[email protected]. † Current address: College of Biological Science and Engineering, Fuzhou University, Fuzhou China.
The classical Hamiltonian for the A + BC f AB + C or AC + B reaction in a CNT is shown in eq 1,
10.1021/jp804464x CCC: $40.75 2008 American Chemical Society Published on Web 09/04/2008
D + H2 f HD + H Confined in Carbon Nanotubes
J. Phys. Chem. C, Vol. 112, No. 39, 2008 15261
n pi2 1 2 1 2 1 2 p + p + p + H) + 2mA A 2mB B 2mC C i)1 2m
∑
Pr(VR, V, j) ) lim
Nf∞
V(rA, rB, rC, r1, r2, ..., rn) (1) where rA, rB, and rC are laboratory position vectors; pA, pB and pC are canonically conjugate momentum vectors; and mA, mB, and mC are the masses of atoms A, B, and C, respectively. We assume n carbon atoms of mass m ) 12 amu are in the CNT with position vectors and momentum vectors ri and pi, i ) 1, 2,..., n. For each collision energy of interest we numerically integrate the corresponding Hamiltonian equations of motion for an ensemble of initial coordinates and momenta. We take the ensemble to correspond to the A atom (deuterium in Section III) moving parallel to the long axis of the CNT, with a translationally stationary BC molecule (H2 in Section III) with internal degrees of freedom consistent with probability distributions of previously determined quantum-confined states.15 To best understand our approach, and also because we compare to gas phase trajectory reaction probabilities, we first briefly outline some details for obtaining the gas phase reaction probabilities (Section IIA). We then discuss the full A + BC@CNT trajectory procedure (Section IIB) and give details of the potential function (V) employed (Section IIC). A. Classical Dynamics of the Isolated System. The initial condition sampling procedure of Karplus, Porter, and Sharma16 is used, so we note just a few relevant features. Jacobi vectors, r and R, and ABC center of mass (COM) vector (D) related to the atomic position vectors via eq 2 are used (M ) mA + mB + mC).
()
(
0
r 1 R ) D mA M
-1 1 -mB -mC mB + mC mB + mC mB mC M M
)
() rA rB rC
(2)
Appropriate relations connecting the corresponding momentum vectorsspr, pR, and pDsto those associated with the laboratory position vectorsspA, pB, pCsare easily written down.16 For this isolated system, the potential does not depend on D so that pD may be taken to be the zero vector. The COM of the diatomic molecule BC is placed at the origin, and the r and pr vectors associated with BC are sampled according to a quasiclassical distribution consistent with BC vibrational and rotational quantum numbers V and j. This involves taking r ) |r| to be at the outer or inner turning point (r() of a vibration associated with BC having energy EVj, where EVj is the quantized diatomic energy. In this case, the initial angular orientation of r is random. Atom A is then positioned relative to BC according to R ) [0, b, -(F2 - b2)]T, which corresponds to impact parameter b, with b being appropriately randomly sampled up to maximum impact parameter bmax, and F is sufficiently large to place the initial condition in the entrance channel.16 The initial BC vibrational phase is accounted for by slight adjustments in the value of F.16 The initial momentum vector for the motion of A relative to BC is pR ) (0, 0, µRVR)T with µR ) mA(mB + mC)/M, with VR being the magnitude of the initial relative velocity. The reaction probability as a function of initial relative speed from an initial ro-vibrational state of the reactant molecule, Pr(VR,V,j), is then:
Nr(VR, V, j) N(VR, V, j)
(3)
where N is the total number of trajectories sampled, and Nr is the number of reactive trajectories. The standard error associated with eq 3 is (Pr[(N - Nr)/(NNr)]1/2. In all calculations, the H2 is initially in its ground vibrational state. The initial H2 rotational state distributions, however, are chosen to be consistent with the particular nanotubes as described in Section IIB below. B. Classical Dynamics of the Confined System. As with the isolated A + BC case in Section IIA, it is most convenient to determine the initial conditions for atoms A, B, and C in the confined system of the CNT in terms of the Jacobi and COM vectorssr, R and Dsand associated momentaspr, pR, and pD. A straightforward transformation to the atomic position and momentum vectors is then made. The n carbon atoms in the CNT are placed at their equilibrium values with zero momentum vectors. We define a space-fixed Cartesian coordinate system with the Z-axis coincident with the long axis of the nanotube. The X- and Y-axes thus lie in the transverse plane. The initial relative speed is VR ) (2E/µR), where E is the collision energy. We place the COM of BC at the origin, which is also taken to be the center of the CNT. Incoming A (i.e., deuterium) atoms are all taken to be lying somewhere in a circular area in the X-Y plane with Z ) -(F2 - b2) and initially moving parallel to the long axis of the nanotube, that is, R ) (b cos η, b sin η, -(F2 - b2) and pR ) (0, 0, µRVR)T. Here, η is a randomly chosen angle, and b and F are as in Section IIA. Unlike the isolated gas phase case, the ABC center of mass momentum vector is no longer a constant, so D and pD need to be specified. With the COM of BC at the origin, it is easy to see that D ) mAR/M, and if we choose the BC center of mass to be initially at rest (pB + pC ) 0), then it is also straightforward to show that pD ) pA ) µApR/µR. Choosing the A atoms to be moving parallel to Z makes the confined calculation somewhat analogous to the isolated case (Section IIA) and is the reason why we chose to do so. Note that other initial orientations for the incoming A atoms could lead to different results owing to the anisotropic nature of the CNT environment. It only remains to determine r and pr associated with BC in the CNT. For the relevant case of BC ) H2 we previously determined the rotational states corresponding to H2 confined to various chirality CNTs.15,17 As in the isolated case, we take the initial vibrational quantum number V ) 0. We select j according to the corresponding spherical harmonic content of the ground-state of each CNT quantum-confined wave function of interest. Except for the (8, 0) nanotube, it turns out that the ground-state wave functions are dominated by j ) 0. For the (8, 0) nanotube however 36% of the probability density is in j ) 2, whereas the rest is in j ) 0. Therefore, we choose a pseudorandom number (ξ) between 0 and 1 and take j ) 0 if ξ > 0.36 and j ) 2 otherwise. (As stated above, we use the same initial j distributions in the corresponding calculations for the isolated system) The magnitude of r and the momentum vector (pr) are then determined exactly as in the isolated case. The angular orientation of r, however, is chosen on the basis of the corresponding ground-state wave function. The wave function model for H2 confined in a CNT of ref 15 involved four coordinatesstwo angular coordinates for orientation and two translational coordinates: ψ(x, y, cos θ, φ). We then form the reduced densities,
15262 J. Phys. Chem. C, Vol. 112, No. 39, 2008
∫ |ψ(x, y, cos θ, φ)|2dx dy dφ Pφ(φ) ) ∫ |ψ(x, y, cos θ, φ)|2dx dy d(cos θ)
Lu et al. TABLE 1: Initial H2 CNT-confined State Probabilities for cos θ Lying in Various Rangesa
Pθ(cos θ) )
(4)
where θ is the angle that the H2 makes with the long axis of the nanotube (Z-axis) and φ′ is an azimuthal angle in the X-Y plane. The angular variables cos θ and φ are sampled randomly from these reduced density distributions. [Let u ) cos θ or φ. If P(u) is a probability density defined on [u1, u2] such that P(u) du is a probability, then we construct a numerical representation of G(u) ) ∫uu1P(u′) du′ and then infer a numerical representation for the inverse function, u(G). If χ is another pseudorandom number between 0 and 1, then u(χ) is a value for the variable u randomly sampled according to the probability distribution.] C. Interaction Potential and Additional Computational Details. Unfortunately, the subtle, dispersive interactions between hydrogen atoms and the carbon atoms in a CNT cannot be reliably predicted without extensive and currently computationally too-demanding high-level electronic structure calculations. Rather than use some level of ab initio electronic structure theory or density functional theory that is likely inadequate, it is better in the present case to adopt a semiempirical approach. The interactions between the nanotube carbon atoms and between the carbon atoms and D or H atoms are described by the ARIEBO potential,18 which is a many-body potential. The AIREBO potential is an extension of the REBO19-21 potential; both are Tersoff-type empirical bond order potentials. Tersofftype potentials have been successfully used to model changes in covalent bonding for many materials.22-26 The reactive empirical bond-order (REBO) potential by Brenner was initially developed for simulation of the chemical vapor deposition of diamond and later extended to model the energetic, elastic, and vibrational properties of solid carbon and small hydrocarbons, including carbon nanotubes, but the absence of dispersion and nonbonded terms makes the REBO potential inappropriate for any system with significant intermolecular interactions. The ARIEBO potential improves the REBO potential by adding dihedral-angle interactions that are absent in REBO potential and with a carefully designed mechanism to switch on or off nonbonded interactions. The intermolecular interactions in the ARIEBO potential are modeled with a Lennard-Jones (L-J) 12-6 potential. The parameters for the C-H interaction in the AIREBO code are σ ) 3.03 Å and ε ) 16.68 cm-1. These two parameters are close to the L-J potential parameters in the Frankland-Brenner potential27 that we used in the quantum calculation: σ ) 3.08Å and ε ) 19.25 cm-1. In light of previous work in which we looked at the effects of the H-C interaction potential on H2 confined in these same nanotubes,15 we believe that our results will depend most sensitively on the value of σ, the range parameter. If it were increased, then the confinement effects in the smaller nanotubes will be reduced. The total potential energy of the full system is expressed in eq 5,
V ) VC-C + VC-H,D + VDHH
(5)
where VC-C is a sum over all C-C ARIEBO-based interactions, VC-D,H is a sum over all C-H or C-D ARIEBO-based interactions, and VDHH is the interaction potential for the DHH system. In principle, the original ARIEBO potential could describe the DHH interaction. However, it is not designed to reproduce the correct potential topology of the D + H2 exchange reaction, which only results from high-level electronic structure theory calculations. We modified the ARIEBO code so that the
(n,m), d (Å) (8, d (2, d (6, d
0), ) 6.26 8), ) 7.2 6), ) 8.14
cos θ ) 0.0-0.33
cos θ ) 0.33-0.67
cos θ ) 0.67-1.0
0.05
0.29
0.66
0.38
0.38
0.24
0.46
0.36
0.18
a Where θ is the angle that H2 makes with the long axis of the nanotube.
VDHH term corresponds to the LSTH potential.28,29 This is a much more accurate potential and is also the potential used in our recent quantum work described in Paper I.5 Note that the many body terms used by the Brenner force field to reflect the hybridization state or coordinate number of an atom is determined only by the positions of its neighboring atoms. Because we only replace the forces between the D and H atoms, the AIREBO potential still recognizes the hybridization state of the D and H atoms, ithat is, whether they are isolated atoms, in a diatomic DH/H2 state or in some intermediate or transition state. Thus, the interactions of these three atoms with the carbon atoms of nanotube will not be affected by using the LSTH potential for the internal interaction among the reactants. The number of atoms, n, for each model ranged from 384 to 528 depending on the size of the nanotube. The carbon nanotube structures were generated using the Tubegen program.30 The lengths of the nanotubes ranged from 51.2 Å for the (8, 0) nanotube to 54.1 Å for the (6, 6) nanotube. Periodic boundary conditions were applied to the two terminals of the nanotubes so that these nanotubes were actually modeled as infinity long straight nanotubes. A sixth-order Adams-Moulton method with a Runge-Kutta start-up was used to integrate the trajectories.31 The time step was 0.05 fs. Each trajectory is propagated for 5000 steps for a total of 0.25 ps. The standard deviation of the total energy during the simulation is about 2 × 10-5 eV. For each case, we computed results for 13 collision energies, from ranging from 0.3 to 1.5 eV. There are seven cases; in addition to the isolated system (model g), for each of the three nanotubes, we computed both nanoconfined trajectories (model A) and an ensemble of trajectories with the same initial conditions as model A but without the nanotube potential (model a). For each case and each collision energy, we computed 6000 trajectories for a total of 7.02 × 105 trajectories for this study. Except for the lowest collision energy 0.3 eV, which has a small probability for reaction, the statistical error for the probabilities is in the range of 1-4%. The largest statistical error overall is 16%. At each collision energy, we used the same maximum impact parameter, bmax, for all models, that is, models A, a, and g. Maximum impact parameters as a function of collision energy are given in Table 2. III. Results and Discussion To see how the initial confined H2 states are oriented relative to the long axis of the nanotube, we integrated Pθ(cos θ), eq 4, over various ranges in cos θ (Table 1). (It turns out that Pφ(φ) is approximately constant and so is less relevant.) We see that for the smallest diameter CNT, (8, 0), that there is a definite preference for H2 to initially be nearly parallel to the long axis of the nanotube (θ ) 0 or cos θ ) 1). As the nanotubes get larger, however, there is a stronger preference for the H2 to be
D + H2 f HD + H Confined in Carbon Nanotubes
J. Phys. Chem. C, Vol. 112, No. 39, 2008 15263
TABLE 2: Maximum Impact Parameters As a Function of Collision Energya collision energy (eV)
bmax (Å)
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5
0.46 0.84 0.99 1.08 1.16 1.20 1.24 1.27 1.29 1.31 1.33 1.34 1.34
a The same bmax value was used for all calculations at a specific collision energy.
Figure 1. Reaction probabilities for models A, a, g in the (8, 0) nanotube.
perpendicular to the nanotube axis (θ ) π/2 or cos θ ) 0). If we consider low impact parameter collisions, reactants in the (8, 0) nanotube are thus more favorably aligned for a reaction such as D + Η2 f DΗ + H that goes through a collinear transition state, and reactants in the (6, 6) nanotube would correspondingly be less favorably aligned. Reaction probabilities for the D + H2 reaction confined in the (8, 0), (2, 8), and (6, 6) nanotubes are shown in Figures 1, 3 and 4. In each of these plots we compare the nanoconfined reaction probabilities (A) with those from the gas phase (g) and from the ensemble of trajectories with the confined state initial conditions but without the nanotube confinement (a). A. The (8, 0) Nanotube. From Figure 1 we see at all collision energies the reaction probabilities are such that model A > model a > model g. This implies that there are two factors contributing to the enhanced reaction probability. The enhanced reactivity of model a over model g, both using gas phase calculations but with model a having initial conditions that mimic confined H2, is entirely due to the favorable orientation of the H2. The enhanced reactivity of model A, the fully confined case, over model a must be due to confinement effects during the course of the collisions. Because of the small diameter of the nanotube, the D atom experiences strong repulsive interactions with the nanotube walls, with attractive forces steering it to the center of the nanotube. (There is a short-range chemical interaction between the D atom and the nanotube carbons, but there is a potential barrier that is high enough so that it is not
Figure 2. Trajectories with the same D + H2 initial conditions. Left panel is without nanotube (model a); right panel is within an (8,0) nanotube (model A). Time progresses from top to bottom in 0.01 ps intervals. The collision energy is 1.0 eV.
relevant at the collision energies used in this study.32) These steering effects increase the likelihood that even large impact parameter trajectories will access the transition state. Additionally, hindered rotation of the confined H2 can contribute to the reaction. In Figure 2, we show side-by-side two trajectories that emanate from the same D + H2 initial conditions. In the trajectory on the left-hand panel, the nanotube is not present (model a); on the right, the system is confined in the (8,0)
15264 J. Phys. Chem. C, Vol. 112, No. 39, 2008
Figure 3. Reaction probabilities for models A, a, g in the (2,8) nanotube.
Figure 4. Reaction probabilities for models A, a, g in the (6, 6) nanotube.
nanotube (model A). The model A trajectory is reactive but the model a trajectory is not. The D atom approaches one of the H atoms so as to cause the H2 to rotate slightly in a way that is not conducive to reaction (t ) 0.03 ps, fourth frame down). Within the nanotube, however, this rotation is hindered, and the H2 rotates toward the D atom. B. The (2, 8) and (6, 6) Nanotubes. Figures 3 and 4 display the reaction probabilities for the (2, 8) and (6, 6) nanotubes, respectively. The (2, 8) results are such that whereas the fully confined model A yields the largest probabilities, the reaction probabilities for model g are greater than those for model a, which is the opposite of the (8, 0) result. In this case, the confined H2 orientations do not favor reaction; in fact, they are less favorable than the random orientations of the isolated system. Nevertheless, the confining influence of the CNT results in an enhanced reaction for the confined reactants, although not as dramatically as in the smaller (8, 0) CNT, as can be seen in Figure 5, where we compare model A for all three nanotubes. On the other hand, for the largest diameter nanotube considered, (6, 6), the reaction probabilities for ordinary gas phase reaction probabilities of model g are the highest (Figure 3). Thus, given the assumptions of our model, confinement in the (6, 6) nanotube actually reduces the reaction probability. Several factors contribute to this reduction. The most important factor is that the initial distribution of H2 orientations for larger nanotubes such as (6, 6) favors perpendicular orientations
Lu et al.
Figure 5. Model A reaction probabilities in the (8, 0), (6, 6), and (2, 8) nanotubes.
relative to the Z-axis as opposed to parallel orientations. Furthermore, in the larger nanotubes, there are no strong forces driving the D atom toward the center of the nanotube. The model presented here, however, is perhaps not the most appropriate for large nanotubes because in these systems the H2 molecule prefers to be closer to the nanotube walls,15 and the COM of the entire system is not on the central axis.5 It is thus possible that with a more general method of choosing initial conditions the reaction probabilities will be enhanced, even in the larger nanotubes. We explore this issue in the next paper in this series. In our calculations, we allow for the motion of the nanotube carbons. This is necessary to properly treat the interactions of D and H2 with the nanotube walls. We also wanted to allow for the possibility that some D atoms might actually bind to the nanotube walls, transferring their kinetic energy to the nanotube carbons. Permitting this motion is especially important in models in which collisions with the walls play an important role, which we discuss elsewhere. In our quantum calculation, however, we fixed the carbon atoms. So, in order to assess the affect of using fixed carbon atoms in a model where the likelihood of collisions with the walls is minimal, we repeated the calculations for the (8,0) nanotube with fixed carbon atoms. The reaction probabilities turn out to be only slightly higher ( 0. Thus, the larger enhancement of the reaction probabilities in the classical calculations is likely due to the fact that we include all impact parameters rather than quantum effects or other differences in the two calculations. In the larger nanotubes, the steering effect does not play the same role for several reasons: (1) the forces steering the D atom toward the center of the nanotube are much weaker, and (2) the initial orientation of the H2 does not favor head-on collisions. In Table 4 we give enhancement factors at specific collision energies. Here it can be seen that in all cases the effects of confinement are more pronounced at lower collision energies. This is true whether the reactivity is enhanced by confinement, as in the smaller nanotubes, or if it is quenched, as in the classical calculation for the (6, 6) nanotube. By comparing the classical reaction probability ratios A/a and a/g with A/g, we can also estimate what fraction of the enhancement of the overall reaction probability comes from the initial conditions and from confinement effects during the course of the trajectory. Table 2 shows that for the (8, 0) nanotube it appears that it is roughly half-and-half. For the (2, 8) nanotube, of course, the initial conditions do not favor reaction, so all of the enhancement comes from the confined trajectory. Finally, for the (6, 6) nanotube, the reduction of
reaction probability in the classical calculations appears to be entirely a result of the unfavorable initial orientation of the hydrogen. IV. Summary We carried out extensive classical trajectory calculations on the D + H2 reaction confined to three different nanotubes. Our results were compared with more standard isolated gas phase reaction probabilities. We found that reaction probabilities are enhanced in the two smallest diameter nanotubes, (8, 0) and (2, 8), and were somewhat quenched in the largest diameter nanotube, (6, 6). Enhancement (or quenching) arises as a result of at least two factors: the preferred initial conditions of the reactants and the confinement effects during the course of the reaction. For the smallest (8, 0) nanotube, both factors work toward favoring an enhanced reaction probability. Confined H2 reactants tend to be oriented along the nanotube axis. Additionally, during the course of the reaction the D atoms are steered toward the center, favoring more reactive, head-on, near-collinear collisions. In the case of the larger (6, 6) nanotube, the opposite situation prevails. The H2 reactants prefer a perpendicular orientation with respect to the nanotube axis, and the D atom prefers to be closer to the nanotube walls. These factors do not favor accessing the transition state. The (2, 8) nanotube is intermediate between the smaller and larger nanotube. The initial conditions do not favor reaction, but confinement during the course of reaction leads to a modest but significant enhancement of reactivity. Our results strongly suggest that confinement effects can be used to enhance or quench chemical reactivity. The reactant initial conditions explored here are, however, somewhat limited. They are consistent with a beam of deuterium atoms injected into the nanotube with a velocity along the long axis colliding with hydrogen molecules centered on the nanotube axis. We have studied more general models that do not have these restrictions and in which collisions with the walls play at important role in enhancing reactivity for all nanotubes, large and small. This latter work will be described in a planned, subsequent publication. Acknowledgment. E.M.G. acknowledges support from the U.S. Department of Energy, Basic Energy Sciences, Grant No. DE-FG02-01ER15212. S.K.G. was supported by the U.S. Department of Energy, Basic Energy Sciences, under contract No. DE-AC02-06CH11357. References and Notes (1) Britz, D. A.; Khlobystov, A. N.; Porfyrakis, K.; Ardavan, A.; Briggs, G. A. D. Chem. Comm. 2005, 37. (2) Kondratyuk, P.; Yates, J. T. J. Am. Chem. Soc. 2007, 129, 8736. (3) Pan, X. L.; Fan, Z. L.; Chen, W.; Ding, Y. J.; Luo, H. Y.; Bao, X. H. Nat. Mater. 2007, 6, 507. (4) Chen, W.; Pan, X. L.; Bao, X. H. J. Am. Chem. Soc. 2007, 129, 7421. (5) Lu, T.; Goldfield, E. M.; Gray, S. K. J. Phys. Chem. C 2008, 112, 2654. (6) Banares, L.; Aoiz, F. J.; Herrero, V. J. Phys. Scr. 2006, 73, C6. (7) Juanes-Marcos, J. C.; Althorpe, S. C. J. Chem. Phys. 2005, 122. (8) Juanes-Marcos, J. C.; Althorpe, S. C.; Wrede, E. Science 2005, 309, 1227. (9) Kendrick, B. K. J. Chem. Phys. 2001, 114, 8796. (10) Kendrick, B. K. J. Chem. Phys. 2001, 114, 4335. (11) Kendrick, B. K.; Jayasinghe, L.; Noser, S.; Auzinsh, M.; ShaferRay, N. Phys. ReV. Lett. 2001, 86, 2482. (12) Aoiz, F. J.; Banares, L.; Herrero, V. J. Int. ReV. Phys. Chem. 2005, 24, 119.
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JP804464X
