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J. Phys. Chem. C 2010, 114, 9030–9040
Classical Trajectory Studies of the D + H2 f HD + H Reaction Confined in Carbon Nanotubes: Effects of Collisions with the Nanotube Walls Tun Lu† and Evelyn M. Goldfield* Department of Chemistry, Wayne State UniVersity, Detroit, Michigan 48202
Stephen K. Gray Center for Nanoscale Materials, Argonne National Laboratory, Argonne, Illinois 60439 ReceiVed: February 28, 2010
We use full-dimensional classical trajectories to study how reaction cross sections for the D + H2 f DH + H reaction are altered when the system is confined to move within various sized carbon nanotubes (CNTs). We focus on trajectories with initial conditions such that collisions with the nanotube walls are possible. Unlike our previous studies where the initial conditions minimized the potential for such collisions [Lu, T.; Goldfield, E. M.; Gray, S. K. J. Phys. Chem. C 2008, 112, 15260], we find that reaction cross sections are enhanced in all the differently sized CNTs compared to cross sections in the isolated systems, although the enhancements are larger for the smaller CNTs. We interpret our results based on a simple specular reflection model for collision cross sections within a cylinder. I. Introduction The effect of confinement in nanoscale media on chemical reactivity has generated recent experimental interest. In particular, it has been shown that confinement of molecules in carbon nanotubes (CNTs) may have considerable effect on their reactivity. For example, Britz et al. have shown that it is possible to produce linear rather than branched fullerene polymers by heating C60O inside single-walled CNTs.1 Kondratyuk and Yates2 demonstrated that 1-heptene molecules inside CNTs are less reactive to atomic hydrogen than 1-heptene molecules adsorbed on external CNT sites. More recently, in a series of remarkable experiments Pan, Bao, and co-workers have shown that the redox properties of transition metal catalysts can be modified by confinement in CNTs thereby tuning their performance as catalysts.3-6 A review article by Pan and Bao3 summarizes both their work and that of others on reactions in CNTs and other nanoporous materials. Several interesting theoretical studies have explored the effects of confinement in CNTs on the chemical reactivity, particularly on the features of the reaction paths.7-9 Recent works by Santiso, Gubbins, and co-workers have explored the confinement effects on reactions in a variety of nanoporous materials.10-13 There have been, however, few studies that focus explicitly on reaction dynamics in CNTs. Studies of the dynamics of simple reactions, such as the reaction of an atom with a diatomic molecule confined to the interior of a nanotube, can uncover important features of confined reactions that may not be revealed by the minimum energy pathway. These simple reactions have been well studied in the gas phase, and it is relatively straightforward to assess the effects of the confining media on their reactivity. This study is our third exploration of the dynamics of D + H2 confined in a CNT, which we refer to as D + H2@CNT. In previous studies,14,15 we explored the quantum dynamics of the * To whom correspondence should be addressed. † Current address, College of Biological Science and Engineering, Fuzhou University, Fuzhou, China.
D + H2 f HD + H reaction confined to occur within CNTs of various sizes and structures, using both quantum (paper I)14 and classical dynamics (paper II).15 Both of these studies were designed to elucidate the effects of physical confinement, i.e., of the reduction of conformational space available to the reactants, on chemical reactivity. In this paper we emphasize a different aspect of physical confinement caused by the incoming D atom directly colliding with the walls of the CNT container. The models employed in the two previous papers minimized the importance of such wall effects since the incoming deuterium atom initially was directed parallel to the long axis of the nanotube. Here we consider two more general models for the initial conditions of the reactants in which collisions with the walls of the nanotube are likely to occur. As in previous work we confine the interacting system in three nanotubes with various chiral vectors (n,m) and associated diameters, d: (8,0), d ) 6.26 Å; (2, 8), d ) 7.2 Å; and (6,6), d ) 8.14 Å. One of the difficulties when attempting to compare the reactivities of confined systems to their unconfined counterparts is that some of the assumptions and concepts generally employed in computing the collision cross section for an isolated system cannot be applied in a straightforward manner for the confined system. Thus in this work we develop an alternative definition for the collision cross section and a simple model for understanding the role of reflections from the walls. It is our hope that this simple model will allow us to compare results between the confined and unconfined reactions in a straightforward manner. In an isolated system, two factors simplify the calculation of the cross section, σr, for a chemical reaction or other process of interest: (1) σr is independent of the orientation of the interacting system in space and (2) as long as the initial separation of the particles, R, is large enough so that the interaction potential between them is negligible, σr is independent of the particular value of R. Neither of these factors is true for a system within a confining media, and our model will
10.1021/jp101808p 2010 American Chemical Society Published on Web 04/19/2010
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Figure 1. Schematic drawing of a conventional circular collision cross section with impact parameter bmax. Trajectories are shown as arrows.
need to take this fact into account. In this work we: (1) develop a simple, specular reflection model for collisions between particles confined within a hard wall cylinder and (2) compute full-dimensional classical trajectories for D + H2@CNT, using methods for selecting initial conditions in which collisions with the nanotube walls will be likely, using the simple model to help interpret results. Section II below discusses and illustrates the cylinder model, and section III discusses our more extensive classical trajectory calculations of D + H2@CNT. Section IV concludes. II. Simple Model for Collisions in a Hard Wall Cylinder A. Cross Section in an Isolated System. We begin the discussion by considering collisions of an isolated system. For collisions between two isolated particles A and B (which may be atoms or molecules), a general cross section as a function of total energy, E, is given by integration over impact parameters, b16,17
σ(E) ) 2π
∫0b
max
P(b)b db
(1)
where bmax is the largest impact parameter for which the process of interest, for example, reactive scattering or energy transfer, occurs. Here the opacity function P(b) subsumes averages over all the other degrees of freedom in the system. The collision cross section is defined asP(b) ) 1 for b e bmax, with bmax being the largest value of b such that there is a “measurable” deflection angle.18 In cross sections for other processes we may have P(b) < 1. Although we are interested in the chemical reaction, much of the following discussion will focus on the collision cross section. In practical trajectory calculations, excluding trajectories such that b > bmax in eq 1 greatly reduces the computing load by ignoring those trajectories that are known to be nonreactive. Note that the initial distance between the two particles does not come into the equation; i.e., it is assumed to be infinite. (In practice a finite, but sufficiently large initial distance, is used.) All trajectories that contribute to σ are contained within a cylinder of radius of bmax, as shown in Figure 1. As long as the interaction potential between the two particles is negligible at the given internuclear separation, however, the particular initial distance used is of no importance. In actual calculations that employ Monte Carlo sampling of initial 2 , conditions, with b2 being evenly sampled between 0 and bmax 16 the reaction cross section is computed as
σr )
Nr 2 πb N max
(2)
where Nr is the number of reactive trajectories in an ensemble of N trajectories.
Figure 2. A schematic illustration of the spherical zone cross section for the collision of A with B. A slice in the xz plane is shown. The large black half circle is the projection in the xz plane of a half-sphere with radius R centered at point A; the blue circle is the projection of a sphere centered at point B with radius bmax. The dashed red lines are the projections of the cone defined by the tangent lines from point A to sphere B(bmax). Only trajectories inside this cone will collide with B. The intersection of the cone with half-sphere (A) forms a spherical zone. The portion of the spherical zone on half-sphere (A) that lines in the xz plane is also colored red. All trajectories emanating from A that can collide with B will intersect this spherical zone. An equivalent picture could be drawn in the yz plane.
In order to study trajectories of a system within a cylindrical cavity, we consider a somewhat different method of defining initial conditions. Referring to Figure 2, we explore a model in which all of the trajectories emanate from a particular point in configuration space. Consider two points, A and B, defined to be the center-of-masses of particles A and B separated by a distance, R. Let sphere A(R) be a sphere with radius R centered at point A. All trajectories that emanate from A will eventually cross the surface of this sphere. Let sphere B(bmax) be a sphere of radius bmax centered on point B. Consider lines from point A that are tangent to sphere B(b). The union of all these lines, which may be thought of as trajectories with impact parameters equal to bmax, form a cone-shaped surface. All trajectories contained inside this cone have impact parameters less than bmax and the set of all trajectories that emanate from point A with b e bmax is represented by a solid cone. The intersection of this solid cone and the surface of sphere A(R) forms a spherical zone on the sphere’s surface. A schematic illustration of a slice of the spherical zone in the xz plane is given in Figure 2, where the z axis contains both point A and point B. The surface area of the spherical zone defined above is given by
S0(bmax) )
θ 2πR2 sin θ dθ ) 2πR2(1 - cos θ0) ∫θ)0 0
(3) where θ0 ) θ0(bmax) is given by
sin θ0 )
bmax R
(4)
It is the area defined by eq 3 and related areas associated with wall collisions that form the basis of the cross section model of
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this paper. Note that eq 4 implies that bmax e R. In this case, we can express S0 as a function of R and bmax
S0 ) 2π(R2 - R√R2 - bmax2)
(5)
The above calculation assumes the z axis is defined by the line connecting A and B in Figure 2 so that either the x or y axis is normal to it and in the plane. Note that when R f ∞, it is easy to show that S0 reduces to the standard cross sectional 2 . area that would result from eq 1, σ(bmax) ) πbmax In addition to the difference in the magnitude of the cross sectional areas for finite R, the standard model and our model also weight trajectories with the same impact parameter differently. To show this, we compare the ratio of the area of two standard cross sectional areas σ(b1) and σ(b2), with b1 < b2, to the ratio of the area of two spherical zone cross sections S0(b1) and S0(b2). Let
X ) σ(b1)/σ(b2) ) b12 /b22
(6)
and
Y ) S0(b1)/S0(b2) ) (1 - cos θ1)/(1 - cos θ2)
(7)
where the θi satisfy: sin θi ) bi/R and 0 e θi e π/2. Using trigonometric identities, we find
X)
b12 b22
θ1 2 )Y θ 2 cos2 2 cos2
(8)
Because θ1 < θ2, we see that X > Y. Thus the cross section S0 gives less weight to small impact parameter trajectories than the conventional cross section. In general, however, collisions with small impact parameters are the most reactive. Thus, all else being equal, the average reaction probability will tend to be smaller if sampling is done inside a cone instead of inside a cylinder. Note that at infinite R, θi ) 0 and X ) Y, as expected. The reader may be concerned that our cross section depends upon the initial value of R that we choose, which is somewhat arbitrary. For example, for R values slightly above bmax, S0 given by eq 5 can be much larger than σ(bmax). Therefore one might be concerned that our model is predisposing the cross section to be large. For the realistic values of R that we employ, however, S0(R) ≈ σ(bmax) Moreover, the particular value of R does have a dynamical significance which we discuss here. Consider all straight line trajectories that emanate from point A in the positive z direction. We can imagine that these trajectories are coming in from infinitely far away, but we consider the initial state to be the moment each trajectory passes through point A, defined by the angles θ and φ and corresponding to a range of impact parameters. It is clear that R defines the largest impact parameter that occurs in the model, corresponding to trajectories that make an angle just infinitesimally under θ ) π/2 with the z axis, i.e., trajectories going almost straight up the x axis correspond to impact parameter b ≈ R. Without wall collisions these extreme trajectories and indeed all trajectories with angles such that they are not within the cone of Figure 2 are not reactive. Allowing for wall
Figure 3. Schematic illustration of isolated trajectories (top) and confined trajectories (bottom). Collisions with the nanotube wall change the initial impact parameter, b0, to b1. In many cases b1 < b0.
collisions they could be reactive, and thus R is the more natural upper limit for an impact parameter in our model. This definition of the collision cross section prepares us to develop the model presented in the next section, that hopefully reflects significant aspects of what would actually occur in a confined environment: (1) that the collision partners would assume many different orientations with respect to the walls of the container; (2) that these orientations would influence the dynamics; and (3) that collisions with the walls could play a significant role in determining the outcomes. B. Collision Cross Section within a Cylinder. We now consider a reactive system confined within a hard wall cylindrical cavity. In our model, straight-line trajectories that hit the wall of the cavity undergo simple specular reflection. In this case trajectories initiated with one impact parameter may result in collisions consistent with a different impact parameter. Figure 3 illustrates how collisions with the walls of the cavity may modify the impact parameter. The top panel of Figure 3 shows a gas phase trajectory with impact parameter b0, while the bottom panel shows the same trajectory colliding with the nanotube wall. The effective impact parameter changes from b0 to b1 after hitting the wall and changing its direction of motion. In order to understand the impact of these reflections on the chemical reactivity of the confined system, we will consider a simple model, schematically illustrated in Figure 4. We define a cylindrical cavity with diameter d using a Cartesian coordinate system in which the z axis coincides with the long axis of the cylinder. We place both particle A and particle B on the z axis. We assume that B is stationary and A approaches B. In Figure 4 we are looking at a projection onto the xz plane. The walls of the cylinder are represented as green lines; sphere B(bmax) is represented by the blue circle; half-sphere A(R) is shown in black; R is the initial distance between particles A and B. We consider straight-line trajectories emanating from point A that, in the absence of the cylinder, would eventually intersect halfsphere A(R), as shown by the dashed lines in Figure 4. Each of
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J. Phys. Chem. C, Vol. 114, No. 19, 2010 9033 We can also write Si as follows
Si ) 4πR2
Figure 4. Schematic diagram illustrating the reflection cross section for particles inside a cylindrical cavity.
these lines is specified by spherical angles (θ, φ) where θ is the angle that the line makes with the z axis and φ represents the azimuthal angle in the xy plane, orthogonal to the plane of the paper. Trajectories for which θ satisfies θ1 < θ < θ2 are contained inside a cone defined by two angles θ1 and θ2. The probability for a random straight line to fall into a specific cone is proportional to the surface area of the intersection of the cone and half-sphere A(R). The portions of these intersections in the xz plane are shown in Figure 4 as red arcs on half-sphere A(R). As shown in Figure 4, the red cone contains all the trajectories that can go directly inside sphere B(bmax). In the isolated system, these trajectories represent the totality of collisions that can possibly result in reaction. In addition to these trajectories, trajectories that hit the wall of the cavity will change their direction of motion. If certain conditions are satisfied, they can also hit sphere B(bmax). For example, as shown in the Figure 4, those trajectories inside the purple cone can hit sphere B(bmax) after one reflection. Those trajectories inside the blue cone can reach sphere B(bmax) after two reflections. Although in the isolated system, the trajectories in these cones have b > bmax, and are thus nonreactive, within the cavity they may be reactive. Therefore confinement effects simply due to reflection can increase the number of potentially reactiVe trajectories and hence change the reaction dynamics. Trajectories may fall into sphere B(bmax) after i ) 1, 2, 3, ..., reflections. For each value of i, the trajectories are contained inside a specific cone defined by a pair of angles [θ1(i), θ2(i)]. The surface areas of the intersections of cone [θ1(i), θ2(i)] with half-sphere A(R) are labeled as S0, S1, ..., as shown in Figure 4. The surface area S0 has already been defined in eqs 3 and 4. The subsequent surface areas, with the assumption of specular reflections, are given by
Si ) 4πR2 sin(Ri) sin(θi)
(9)
with
sin(Ri) )
id , Li
Li ) √R2 + (id)2
(10)
and
bmax cos(Ri) sin(θi) ) R
(11)
bmax(id) R2 + (id)2
(12)
The derivation of these equations is given in the Appendix. Note that these equations are true for i g 1; S0 is independent of the diameter of the cylinder. Our interest is to determine a cross-sectional area for this simple model, which would naively correspond to the sum of all Si. With specular wall collisions, there can of course be any number, i, of wall collisions. (As θ approaches π/2, i approaches infinity.) However, in practice, for i > nmax with nmax a relatively small number, the cross sectional areas, Si, for the ever increasing number of wall collisions overlap. It is therefore not meaningful to sum over all i. To obtain a meaningful total reflection cross section, we simply take the angular region of the i ) nmax section to encompass all these other multiple wall collisions; i.e., we take θ2(nmax) ) π/2. This allows us to define the total cross section, ST, as nmax
ST )
∑ Si
(13)
i)0
For fixed d and bmax, ST is a monotonically increasing function of R. This increase is due to the monotonic increase of the surface area of half-sphere A(R) as R increases. Note that as R increases, so does nmax so that more terms contribute to the sum in eq 13. A more interesting quantity than ST itself is the enhancement of the collision cross section due to reflections which we call R, which is given by the ratio of the total reflection collision cross section ST to the collision cross section in the absence of reflection S0; i.e., R ) ST/S0. We plot R as a function of R in the upper panel of Figure 5 for several values of bmax. For example, when bmax ) 0.46 Å and R ) 20 Å, R ) 1613. This means that a random trajectory inside the cylinder is 1613 times more likely to hit sphere B(bmax) than in the isolated system when the initial separation is 20 Å. Note that R is also a monotonically increasing function of R, illustrating the importance of the particular internuclear distance when particle A first emanates from the z axis in the direction of particle B. Also note that smaller values of bmax give rise to larger values of R. Thus the amplification for small bmax is larger than for large bmax. The reflection effects can be viewed as a result of increasing the effective pressure of the reactants. So it might be no surprise to see that reflection increases the probability for collisions to occur. But the fact that this enhancement effect is greater for systems with smaller bmax than for systems with larger bmax is very interesting as it means that, all else being equal, less reactive processes receive a greater “benefit” from the reflections. Note that the enhancement effect is independent of all the particular characteristics of the system except for the maximum impact parameter of the reaction. We define a quantity β, the probability for a random trajectory emanating from the z axis in the direction of particle B to hit sphere B(bmax)
β)
ST 2πR2
(14)
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Lu et al. TABLE 1: Properties of an Idealized (8,0) Nanotube (R ) 4.23 Å) collision energy (eV) bmax (Å) ST (Å2) ST/2πR2 (R) ST/S0 (β) S0/2πR2 St2/S0 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
45.32 65.59 71.35 74.88 78.06 79.67 81.29 82.51 83.26 83.65 84.05 84.24 84.24
0.40 0.58 0.63 0.67 0.69 0.71 0.72 0.73 0.74 0.74 0.75 0.75 0.75
67.97 29.30 22.85 20.10 18.11 17.25 16.46 15.91 15.55 15.14 14.74 14.55 14.55
0.006 0.02 0.028 0.033 0.038 0.041 0.044 0.046 0.048 0.049 0.051 0.051 0.051
29.08 16.27 13.90 12.80 11.95 11.57 11.22 10.97 10.80 10.65 10.49 10.42 10.42
random cos θ and φ. (It is important to note that particle B is simply a reference point; there is no actual representation of a hydrogen molecule.) A trajectory is counted as a hit if during the course of the propagation the distance between the D atom and particle B is less than bmax, i.e., falls within sphere B(bmax). The collision cross section is given by
Nhit (2πR2) Ntot
Figure 5. (top) The enhancement of the collision cross section, given by R ) ST/S0, due to reflections from the walls as a function of the initial separation of the particles for a hard wall (8,0) “nanotube”. (bottom) β ) ST/2πR2 as a function of the initial separation of the particles for the hard wall (8,0) “nanotube”. (See text.)
In the lower panel of Figure 5 we plot β as a function of R for a hard wall (8,0) “nanotube” for R > 2 Å. The figure reveals that β does not vary strongly with R. Note also that unlike R, systems with larger bmax give rise to larger values of β, which makes sense since one would expect that the proportion of halfsphere (A) that is available for reaction would increase with bmax. This cylindrical model can be extended to treat several other cases:19 (1) the particle B is not on the long axis of the cylinder, (2) the cylinder has thick walls, and (3) there are finite repulsive potential interactions between the particles and the walls. The simple model presented here, however, suffices for the purposes of the current study. The properties of an idealized (8,0) nanotube as predicted by this model are given in Table 1. C. Evaluation of the Simple Cylindrical Cavity Model. To test if our simple model can be used to approximate collision dynamics inside a more realistic model of a carbon nanotube, we use classical trajectories to compute the collision cross section of a D atom with sphere B(bmax) inside an (8, 0) carbon nanotube with length of 50 Å. The carbon atoms of the CNT are assumed to be fixed during the simulation. The potential between the carbon atoms of the nanotube and the D atom is given by the ARIEBO potential20 which is an extension of the well-known Brenner potential21-23 as discussed in detail in paper II.15 Both the D atom and particle B (representing the centerof-mass of the H2) are placed on the long axis of the nanotube with R ) 4.23 Å. The initial kinetic energy of the D atom is set to 5.4 eV. The D atom is then projected toward B with
(15)
where Nhit/Ntot is the fraction of trajectories that hit their target. This cross section can be directly compared to the value predicted by our model; this is simply given by ST. As in paper I, each trajectory is integrated using a sixth order AdamsMoulton procedure for 250 fs with 0.05 fs time steps. A total of 3000 trajectories is employed. Figure 6 shows that the agreement between these trajectory results and the simple model is remarkably good. Note that these collision cross sections are much larger than the standard πb2max , suggesting that, due to collisions with the walls, initial impact parameters greater than bmax contribute to the collision cross section. This further suggests that enhanced reactivity within confined spaces may be achieved not only by optimal alignment of reactants but also by simply enlarging the initial configuration space that is available for reaction.
Figure 6. Collision cross sections representing collisions of the D atom with particle B as predicted by trajectory calculations and the simple cylindrical cavity model.
Collisions with the Nanotube Walls III. D + H2 Reaction Dynamics within CNTs Using quasi-classical trajectories, we study the reaction dynamics of D + H2 confined in the (8, 0), (2, 8), and (6, 6) nanotubes, comparing with the dynamics of the isolated system, and keeping in mind the results of section II. Regarding all three confined models A, B, and C discussed below, the computational details are presented in paper II15 and will not repeated here except to note that (1) in these calculations we treat the full dimensionality of the system so that we do not hold the nanotube carbons rigid, (2) the interactions between the nanotube carbon atoms and between the carbon atoms and D or H atoms are described by the ARIEBO potential,20 (3) the forces between the H-H and the D-H atoms are derived from the ab initio LSTH potential,24,25 (4) the initial orientation of the H2 molecule is sampled from the ground state wave function of H2 confined in that specific nanotube,26,27 and (5) H2 is in its ground vibrational state. For all of the nanotubes but the smallest (8, 0) nanotube, the hydrogen molecule is in its rotational ground state. For the (8, 0) nanotube, approximately 36% of the probability is chosen to be in H2 rotational state J ) 2, to reflect the rotational distribution of the wave function of the ground state of para hydrogen confined to an (8, 0) nanotube. We used three different models, models A, B, and C to explore the reaction dynamics of the confined system. These models differ only in the specification of the initial conditions. For each model, we compute a corresponding “gas phase” ensemble, denoted a, b, and c, with the identical initial conditions but without the nanotubes. We also compute model g, which is the standard quasi-classical calculation for an isolated gas-phase system. For each model and each nanotube, we computed results for 13 collision energies, ranging from 0.3 to 1.5 eV. In each case we compute 6000 trajectories. A. Definition of Models A, B, and C and Gas-Phase Analogues. In model A, the center of mass of the H2 is placed on the long axis (z) of the nanotube and the initial velocity of the D atom is directed parallel to that axis. We explored the dynamics of this model in paper II. The initial velocity of the center of mass of the H2 molecule is zero. Thus the impact parameter measures the perpendicular distance of the D atom from the z axis and is chosen randomly as b2 from (0,b2max ) with bmax taken from the gas phase value. The identical set of initial conditions was computed without including the nanotube, i.e., the “gas-phase analogue” model a. The primary difference between model a and a standard gas-phase quasi-classical simulation, model g, lies in the initial orientation of the H2 molecule, with model a corresponding to a confined (hindered) H2 rotor as outlined above and in paper II, whereas model g represents the H2 molecule as a free rotor. In model B, the initial states of H2 are the same as those in model A. The initial position of the D atom, however, is on the z axis. The initial velocity of the D atom is not parallel to the Z axis, as it is in model a, but rather randomly points toward the H2 molecule making an angle θ with the z axis and φ with the X axis, which, of course, is perpendicular to the long axis of the nanotube. This model is therefore closer in spirit to the wall collision model of section II. For trajectories with large initial θ, the attacking atom may collide with the nanotube wall before it approaches the H2 molecule. After being reflected from the wall, the D atom may collide with the H2 molecule or go through more reflections inside the nanotube. Model b is the gas phase analogue of model B; the trajectories have identical initial conditions to those in those in model B, but the nanotube is not present. Thus, this ensemble will have many trajectories with b > bmax.
J. Phys. Chem. C, Vol. 114, No. 19, 2010 9035 Model C is designed to be a more general variation on model B. The major changes from model B are (1) the initial position of H2 is not restricted to be at the center of nanotube, (2) the center of mass of the H2 is not held fixed, and (3) the initial position of the D atom is not fixed at the center of the nanotube but is sampled randomly from the radial profile of the ground state wave function14 for H2@CNT be a distance F from the z axis. (Thus, hindered motions due to confinement of D relative to H2 are also accounted for.) Therefore, in choosing the initial coordinates for the above H2 molecule we use not only the ground state wave function to determine the initial orientation and rotational distribution but also the distance of the center of mass of H2 from the z axis. The initial translational kinetic energy assigned to the H2 molecule is Et ) E0 - Er - V, where E0 is the ground-state energy of H2 inside the nanotube,26,27 V is the potential energy of H2 in its current position, and Er is the rotational kinetic energy. The initial velocity for the center of mass of H2 is restricted to be in the plane perpendicular to the nanotube Z axis but its directions are randomized. In setting the initial conditions for the D atom, we want to be able to compare our results for with those of the other models, particularly model B. But the D atom feels different forces from the nanotube than if it were on the z axis. Therefore, we modified the manner in which we set the initial momentum of the D atom in model C to take into account the potential differences. In models A and B, with a static H2 center of mass, it is easy to see that the initial kinetic energy of the D atom is twice the center of mass collision energy, E. To be consistent with this, in model C we set the initial kinetic energy of the D so that E Ckinetic ) E Bkinetic - (V(F) - V(0)), where the magnitude of E Bkinetic is twice the chosen collision energy. Of course to make the value of kinetic energy meaningful, positions with negative kinetic energy are discarded, which in practice limits the permitted values of F. The direction of the momentum was randomly chosen using the same method as in model B. Note that because of the way that the initial conditions of both the D atom and the H2 center of mass are chosen, E is no longer the collision energy. But when we plot the results for model C as a function of the collision energy that is used to determine Ekinetic, the comparison with model B works out very well. The initial ensemble of trajectories for model c is identical to that of model C and, like model b, contains many trajectories with b > bmax. B. Reaction Cross Sections. In the sections that follow we present reaction cross sections from each of the models. In previous work we compared reaction probabilities. Because the different models use different sampling methods, however, a direct comparison of reaction probabilities is not particularly meaningful. To properly compare, we must account for the different sampling spaces in the various models. For models A, a, and g, the “sampling space” is given by πb2max and the reaction cross section is computed in the standard manner as given by eq 2. Models B and C and their gas phase analogues, however, each have a sampling space of 2πR2 where R is the initial distance between the D atom and the H2 center of mass. Recall that these models sample trajectories with initial impact parameters b e R, so that the larger R is, the larger the range of sampled impact parameters greater than bmax. Thus for these models, the reaction cross section is given by
σ(E) ) P(E)2πR2
(16)
where P(E) is the reaction probability. Using the values for bmax given in Table 1 and R ) 4.22 Å, we can compute these cross sections.
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Figure 8. Comparison of the approach of the D atom to the H2 in models a (red line) and b (green line).
Figure 7. D + H2 reaction cross sections from gas-phase models a, b, c, and g. Models a, b, and c use initial conditions appropriate to the (8,0) CNT (see text).
In addition to the reaction cross sections, we may also compute an idealized cross section that would be predicted from our simple reflection model. We recall that due to reflections, the size of the collision cross sections in which reaction can occur is greater when reflections are possible; in fact it is given by an equation similar to eq 13 n
Stn )
∑ Si
(17)
i)0
where n is the maximum number of reflections that can occur giVen the total integration time and Si is the Ith contribution to the total collision cross section. In our case, we did not have more than two wall collisions so that n ) 2. We scale the model b (c) reaction cross sections so that
* σb(c) ) σb(c)
St2 S0
(18)
Equation 18 gives the predicted reaction cross sections for the confinement in the idealized hard-sphere nanotubes. The values of St2/S0 and several other reflection properties of the idealized (8,0) nanotube described in section IV are listed in Table 1. Note that these Si values are obtained from the simple hard wall cylinder model of section IV, so comparisons based on such scaling can only be used for qualitative purposes. Nevertheless, we believe these comparisons are very instructive. In the following for each CNT, we first discuss the results of the “gas phase” (a,b,c,g) models that differ only with respect to the initial conditions. This comparison allows us to isolate the effects of the initial conditions from the effects of confinement during the course of the trajectory on the subsequent reactivity. We then compare the cross sections for the confined reactions: models A, B, and C. Finally, we compare the model B cross sections with those predicted by hard sphere model. Because our previous studies have shown that confinement effects are much stronger for the smaller (8,0) CNT, we present these results separately from the other two cases. (8,0) CNT. The cross sections for the gas-phase analogues are shown in Figure 7. As discussed in paper II,15 the enhanced reactivity of model a over model g is due entirely to the favorable initial orientation of H2 confined in the (8,0) nanotube. Models a and b use the same method for specifying the initial
conditions of the H2 molecule, and thus one might expect that the reaction cross sections would be equal. But as we can see from Figure 7, the model b cross sections are larger. We explain this as follows: The trajectories of model b are inside the cone depicted in Figure 2 while trajectories of model a are inside a cylinder with radius equal to bmax. For both models, the H-H bond is initially preferentially aligned along the long axis of the nanotube. The direction of an incoming trajectory has a significant impact on the reaction probability. In Figure 8, we draw two trajectories with the same impact parameter b, representing models a and b, respectively. We draw a circle with arbitrary radius with the leftmost H atom of H2 as its center. As seen in Figure 8 this circle crosses the model b trajectory (shown in green) at point D1 and the model a trajectory (shown in red) at D2. Although D1 and D2 are at the same distance from the center of the circle (representing the target H atom), it is always the case that the angle between the incipient D1-H bond and the H-H bond is larger (and thus more collinear) than the angle between the incipient D2-H bond and the H-H bond. Thus, because the linear conformation of D-H-H is more favorable for reaction in the energy region we consider, the model b cross sections are higher than those from model a. From Figure 7 we note that the model c cross sections are similar to the gas phase (model g) at low energy and approach those of model b as the energy increases. In model c, the initial motion of the H2 dampens to some extent the orientation advantage of its initial alignment along the Z axis With higher collision energy or higher relative velocity, it takes the reactants shorter time to reach the transition state region and, thus, the H2 molecule has less time to change its initial alignment. Therefore, the results from model c at higher relative collision energy approach those of model b where the H2 does not move initially at all. In Figure 9 we compare the cross sections for the three models A, B, and C confined to the (8,0) CNT. Several things are immediately obvious. Comparing Figures 7 and 9, we see that the model B and C cross sections are much larger than their gas phase analogues. For collision energies between 0.4 and 1.5 eV, model B cross sections are an average of 7 times greater than its gas phase analogue (model b) and 11 times greater than standard gas phase cross sections (model g). (Model C gives similar results.) Compared to this, model A gives much more modest enhancements: 1.3 (A/a) and 1.9 (A/g). The large enhancement in reactivity is mainly due to reflections from the wall enabling trajectories with initial impact parameter greater than bmax to contribute to the reaction, as discussed above in section II and several other reflection properties of the idealized (8,0) nanotube described in section IV are listed in Table 1. We also see that in the low energy region, the model C cross sections are higher than those of model B, whereas in the higher energy region they are lower and approximately parallel to the
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Figure 9. D + H2 reaction cross sections for the system confined in the (8,0) CNT, models A, B, and C.
Figure 10. D + H2 reaction cross sections for models B, b, and b scaled according to eq 18 as described in text for the (8,0) CNT.
Figure 11. D + H2 reaction cross sections from gas phase models a, b, c, and g. Models a, b, and c use initial conditions appropriate for the (2,8) and (6,6) CNTs, respectively.
results from model B. To explain this, we need to consider two factors with opposite effects: the collision energy and the probability that the D and H2 will collide. Recall that the center of mass H2 in model B is initially static and on the Z axis, but in model C it may be off axis and moving. Allowing the center of mass H2 to move away from the Z axis lowers the probability of collision because the potential energy of the system is lowest at the center of the (8,0) nanotube. This explains the lower reactivity of model C compared to model B for most ranges of the energy. In contrast, at lower D atom energies, the insufficient collision energy becomes the major factor that limits the reaction probability. In this case, the kinetic energy of the H2 can contribute to the total collision energy and enhance the reactivity of model C trajectories over those of model B. In Figure 10, we compare reaction cross sections for model B with both the scaled (as given by eq 18) and unscaled gasphase analogue. The scaled results, the predicted cross sections from the simple hard wall cylinder model of section IV, are considerably higher than the model B results. (Again, results for model C are similar.) This discrepancy is largely due to the fact that the reactivity of the reflected D atoms is less than the reactivity of nonreflected atoms for H2 molecules that are aligned parallel to the long axis of the nanotube. The scaled results assume that the fraction of reactive trajectories contained in S1 and S2 are identical to the fraction of reactive trajectories contained in S0. However, this turns out not to be true. The reflected atoms come from the direction of the wall and tend to attack H2 from the side and thus have lower overall reaction probability than nonreflecting trajectories which hit H2 more head on.
(2,8) and (6,6) CNTs. Reaction cross sections of the gasphase analogues as a function of collision energy using initial conditions appropriate for the (2,8) and (6,6) nanotubes are displayed in Figure 11. In each of these cases, model g has the largest cross section. The diameters of the (2,8) and (6,6) nanotubes are larger than that of the (8,0) nanotube, and we expect that confinement effects will be weaker. As discussed in detail in paper II,15 in both the (2,8) nanotube and (6,6) nanotube, the reaction probabilities from model a are lower than those from model g, just the opposite of the (8,0) nanotube result. This can be explained by the different initial alignment of the H2 molecule inside each nanotube. In the larger nanotubes, the H2 is no longer constrained to lie nearly parallel to the z axis, but in fact, as detailed in papers I and II,14,15 has a broader range of orientations with a preference for being perpendicular to this axis. This alignment does not favor reactivity when the incoming D atom is traveling parallel to the z axis. In contrast to the (8,0) nanotube, the models b and c cross sections are somewhat lower than those of model a. This is what one might expect given that (1) the H2 alignment does not favor models b and c and (2) the cone sampling tends to give less weight to the more reactive small impact trajectories. (See discussion in section II-A.) Figure 12 shows the reaction cross sections from models A, B, and C in the (2,8) and (6,6), respectively. Comparing to Figure 11, we see that the model B and C cross sections are much larger than their gas-phase analogues, although smaller than those for the (8,0) CNT. For the (2,8) CNTs and collision energies between 0.4 and 1.5 eV, model B cross sections are an average of 9 times greater than its gas phase analogue (model
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Figure 12. D + H2 reaction cross sections for the system confined in the (2,8) and (6,6) CNTs, models A, B, and C.
Figure 13. D + H2 reaction cross sections for models B, b, and b scaled according to eq 18 as described in text for (2,8) and (6,6) CNTs.
b) and 6 times greater than standard gas-phase cross sections (model g). The ratios are somewhat less for the (6,6) CNT: 7 (B/b) and 4 (B/g). (Model C results are similar.) Again model A gives modest enhancements for the (2,8) CNT 1.5 (model a) and 1.3 (model g) and no enhancement at all for the (6,6) CNT. In fact, for the (6,6) CNT we have A/g < 1. Thus, collisions with the nanotube walls play a large role in enhancing reactivity even in the larger CNTs. In Figure 13, we compare reaction cross sections for model B with both the scaled and unscaled gas-phase analogue. For the larger nanotubes, these scaled cross sections are much closer to model B cross sections than they are for the (8,0) nanotube. This indicates that for the larger nanotubes, there is not so much difference in reactivity between the reflected and nonreflected trajectories. This agrees with the fact that in the large nanotube, the preferred conformational space of the H2 is not so restricted.
several different models (A, B, and C) for possible initial conditions. We identified three possible confinement effects which enhance the reaction cross section of D + H2 inside small nanotubes: the restricted conformational space of the separated reactants, the restricted conformational space available to the interacting system, and contribution of reflected trajectories. By our design, model a measures the first effect, models A, B, and C measure in different ways the second, and models B and C measure the third. Alignment effects on reactivity have long interested theoretical chemists.28-32 Our results show that size of the nanotube or other confining environment may have a large effect on reactivity due simply to the preferred alignment of the reactants in the particular cavity. In the reaction studied here, for small nanotubes, the H2 prefers to align parallel to the long axis nanotube and in larger nanotube H2 prefers to align perpendicular to this axis. Consequently, the reaction probability of model a, where the incoming D atom is also aligned parallel to the long axis of the nanotube, increased an average of 42% compared to the unconfined gas phase result simply due to this alignment of reactants. For larger (2,8) and (6,6) nanotubes, where the initial alignment of H2 is not favorable for the reaction, reaction probabilities from model a are smaller than those from the gas phase where the H2 can attain all orientations. Confinement within the nanotube affects the entire interacting system throughout the course of the reaction. The conformational space available to the confined transition state, geometrically larger than either reactant, may be different from that of the reactants. Thus in nanotubes where the reactants are not favorably aligned for reaction, there may still exist significant
IV. Concluding Remarks We studied the effect of wall collisions for chemical reactions taking place in confined environments. While our detailed discussion concerned the simple D + H2 f HD + D reaction taking place within several different carbon nanotubes, our considerations should apply to a variety of other reactions. We first developed a simple model for determining effective cross sections assuming specular reflections of reactants within a rigid cylinder (section II). With the aid of the basic concepts and results inferred from this simple model, we then considered fulldimensional classical trajectory calculations (section III). Section III presented results for classical reaction probabilities of the D + H2 system confined inside carbon nanotubes using
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J. Phys. Chem. C, Vol. 114, No. 19, 2010 9039 think of these lines as trajectories that will be reflected at the wall of the cylinder. The reflected trajectories will be tangent lines to sphere B. To draw lines that will be tangent to sphere B after two reflections from the wall, we insert another mirror M2 behind M1. The image of sphere B in M2 will be sphere B′′ with a distance 2d to B. From point A we draw lines tangent to sphere B′′, these lines will be tangent to sphere B after two reflections. This idea may be extended to find tangent lines to sphere B after n reflections. The coordinates of particle B on the cross sectional plane are given by (R,0); thus, the coordinates for its nth reflection, Bn, are (R,nD). Knowing the position of the ith mirror image of particle B, Bi, we can calculate Si. To compute Si we refer to Figure 14b. Here, we use a cylindrical coordinate system (F, φ, z), where, as usual, the z axis is coincident with the cylinder’s long axis, F is the distance to the Z axis, and φ is the angle describing rotation around the z axis with φ ) 0 in the plane of the paper. We assume that particle A is initially at the origin. We draw a half-sphere (A) with radius R. The sphere (Bi) with radius b ) bmax is centered at point (iD, 0, R). Let R be the angle between the Z axis and the line connecting point A and the center of Bi with coordinates (iD,0,R). We have the following relationships
li ) √R2 + (id)2 Figure 14. Calculation of reflection cross section.
confinement effect on the transition state, as we saw with the (2,8) nanotube. Thus, it may be that confinement in semirigid nanocavities that are similar in size to the reactants will have an effect on reactivity that is independent of the particular system and of the particular interactions of the system with the cavity itself. While all three effects play a role, for realistic systems it may well be that the most important confinement effect for enhancing reactivity is reflections from the nanotube walls, particularly in dilute systems, where much of the interior cavity is empty, particle motion is not overly restricted, and the reacting species can collide. This effect will hold for all containers and contribute to enhancing reactivity for all reactions. Compared to a classical container, the nanoscale container is special in that its size is comparable to the length of the mean free path of the reactants. This predicted enhancement contrasts with the reduction in reactivity seen by Kondratyuk and Yates2 where steric shielding of reactants in the nanotube interior played a dominant role. We might expect this to occur when the interior of the confining space is saturated with reactants. We welcome experiments that would determine under which conditions, if any, our predictions of enhanced reactivity will prevail. Acknowledgment. E.M.G. acknowledges support from the U.S. Department of Energy, Basic Energy Sciences, Grant No. DE-FG02-01ER15212. Use of the Center for Nanoscale Materials was supported by the U.S. Department of Energy, Office of Science, Office of Basic Energy Sciences, under Contract No. DE-AC02-06CH11357. Appendix We focus on a cross section of the cylinder that includes the long axis of the cylinder axis, z. As shown in Figure 14, we place the x axis in the plane of the paper and particle A at the origin. We imagine that the wall of the cylinder is a mirror, M1, so that sphere B will have an image sphere B. From point A, we draw tangent lines of length L1 to sphere (B′). We may
cos(Ri) )
R li
for i ) 1, 2, ..., n. We draw two lines tangent to sphere Bi in the plane φ ) 0; these lines make the angles Ri + θi and Ri θi with the Z axis, where
sin(θi) )
bmax li
All of the lines starting from A with angles within the range of (Ri + θi, Ri - θi) can reach sphere (B) after i reflections. Integrating over φ from 0 to 2π, the area of the spherical zone Si is then
Si ) 4πR2 sin(Ri) sin(θi) References and Notes (1) Britz, D. A.; Khlobystov, A. N.; Porfyrakis, K.; Ardavan, A.; Briggs, G. A. D. Chem. Commun. 2005, 37. (2) Kondratyuk, P.; Yates, J. T. J. Am. Chem. Soc. 2007, 129, 8736. (3) Pan, X. L.; Bao, X. H. Chem. Commun. 2008, 6271. (4) Chen, W.; Fan, Z. L.; Pan, X. L.; Bao, X. H. J. Am. Chem. Soc. 2008, 130, 9414. (5) Pan, X. L.; Fan, Z. L.; Chen, W.; Ding, Y. J.; Luo, H. Y.; Bao, X. H. Nat. Mater. 2007, 6, 507. (6) Chen, W.; Pan, X. L.; Bao, X. H. J. Am. Chem. Soc. 2007, 129, 7421. (7) Halls, M. D.; Schlegel, H. B. J. Phys. Chem. B 2002, 106, 1921. (8) Halls, M. D.; Raghavachari, K. Nano Lett. 2005, 5, 1861. (9) Trzaskowski, B.; Adamowicz, L. Theor. Chem. Acc. 2009, 124, 95. (10) Turner, C. H.; Brennan, J. K.; Johnson, J. K.; Gubbins, K. E. J. Chem. Phys. 2002, 116, 2138. (11) Huang, L. P.; Santiso, E. E.; Nardelli, M. B.; Gubbins, K. E. J. Chem. Phys. 2008, 128. (12) Santiso, E. E.; George, A. M.; Gubbins, K. E.; Nardelli, M. B. J. Chem. Phys. 2006, 125. (13) Santiso, E. E.; Nardelli, M. B.; Gubbins, K. E. J. Chem. Phys. 2008, 128.
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(14) Lu, T.; Goldfield, E. M.; Gray, S. K. J. Phys. Chem. C 2008, 112, 2654. (15) Lu, T.; Goldfield, E. M.; Gray, S. K. J. Phys. Chem. C 2008, 112, 15260. (16) Karpus, M.; Porter, R. M.; Sharma, S. C. J. Chem. Phys. 1965, 43, 3259. (17) Atom-Molecule Collision Theory: A Guide to the Experimentalist; Bernstein, R. B. , Ed.; Plenum: New York, 1979. (18) Levine, R. D.; Bernstein, R. B. Molecular Reaction Dynamics and Chemical ReactiVity; Oxford University Press: Oxford, 1987. (19) Lu, T. Theoretical study of D + H2 reaction inside single wall carbon nanotubes. Ph.D. Thesis, Wayne State University, 2007. (20) Stuart, S. J.; Tutein, A. B.; Harrison, J. A. J. Chem. Phys. 2000, 112, 6472. (21) Brenner, D. W.; Garrison, B. J. AdV. Chem. Phys. 1989, 76, 281. (22) Brenner, D. W. Phys. ReV. B 1990, 42, 9458. (23) Brenner, D. W. Carbon 1990, 28, 769.
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