J. Phys. Chem. C 2009, 113, 16945–16950
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Kinetics of External Adsorption on Nanotube Bundles: Surface Heterogeneity Effects Jared T. Burde, Nayeli Zuniga-Hansen, Chong L. Park, and M. Mercedes Calbi* Department of Physics, Southern Illinois UniVersity, Carbondale, Illinois 62901-4401 ReceiVed: June 14, 2009; ReVised Manuscript ReceiVed: August 23, 2009
We investigate the kinetics of adsorption of gases on the external surfaces of a carbon nanotube bundle by simulating the elemental processes that take place during equilibration in a kinetic Monte Carlo scheme. By modeling the adsorbing surface as a lattice composed by three or more linear chains of sites characterized by different binding energies, we are able to describe the overall kinetic behavior of the system as well as to identify the processes responsible for the observed behavior. The presence of sites with different binding energies introduces specific paths of adsorption, as particles use the weaker binding sites as temporary intermediate states before being adsorbed on stronger binding sites. This mechanism greatly increases the adsorption rate of the stronger sites with two main consequences as compared with our previous results for homogeneous surfaces: (1) the time evolution of the coverage is nonexponential, and (2) the overall equilibration time as a function of coverage deviates from the linear dependence. Introduction Single-walled carbon nanotubes exhibit a unique structure that gives them special properties as adsorbents. As they grow, they naturally arrange themselves in a triangular lattice because of the van der Waals forces between them.1 In the case of nanotubes that have been opened, the interiors of the tubes, along with the interstitial channels that exist between adjacent tubes inside the bundle, form a honeycomb-like structure. While this internal structure of the bundle is in general characterized by a pore size distribution, the junction of two neighboring nanotubes on the outside of the bundle forms a groove, creating a corrugated exterior surface. The confinement of particles adsorbed on these internal or external surfaces produces many of the interesting phenomena observed in carbon nanotube bundles.2–4 Particles adsorbed in the pores and grooves interact with all nearby nanotubes and therefore feel different binding energies. They are also limited in movement by the carbon walls, which leads to the development of quasi-one-dimensional adsorption phases, inside both the nanotubes and interstitial channels, and the spreading of similar phases from the groove to form twodimensional films that cover the outside surface of the bundle.3,4 Most adsorption studies of carbon nanotube bundles have focused primarily on the equilibrium between the external gas and the adsorbed phases that characterizes their formation. While this work is critical to understanding the adsorption ability of carbon nanotubes and has produced some very interesting results,2–4 their scope is limited to the final equilibrium configuration of the system. Comparatively, little is known about the kinetics of adsorption, which depends on the path that the system has to follow to reach that equilibrium. Except for very recent studies of the isothermal adsorption kinetics on nanotube bundles,5–8 most of the adsorption kinetics behavior for this system has been derived from temperature programmed desorption experiments9 and studies of molecular diffusion in the bundles.10 The increased need to investigate the kinetics of gas adsorption on nanotube bundles started to be rather evident when discrepancies between experimental results and theoretical * To whom correspondence should be addressed. E-mail: mcalbi@ physics.siu.edu.
predictions of the adsorption behavior began to arise.11 The key issue in this problem comes from the fact that different adsorption sites in a bundle are expected to show a variety of kinetic behaviors, characterized by dramatically different adsorption rates. Understanding this phenomenon is very useful when performing isotherm measurements; in those experiments, a sufficient amount of time should be allowed for the system to reach equilibrium, especially if the gas is expected to intercalate inside the bundle. In addition, adsorption kinetics studies are the first step in assessing the performance on these nanostructures for gas separation and purification applications.12 In a recent work,6 we looked at the dependence of the adsorption rate on the temperature and binding energy, by considering how the equilibration time for adsorption changes as the system approached monolayer completion. In that case, we implemented a kinetic Monte Carlo algorithm (KMC)13 to simulate the adsorption kinetics on linear chains of adsorption sites. When the sites were directly exposed to the gas (as happens in the formation of external phases), and were all characterized by the same binding energy, we observed a linear decrease in the equilibration time of the system as the coverage increases, reaching values close to zero at monolayer completion. The slopes of these lines were observed to increase drastically as the ratio between the binding energy and the temperature was raised.6 That means that adsorption on stronger binding sites (for a given coverage and temperature) would take longer. This is essentially due to the fact that the adsorption rate on a surface directly exposed to the gas is proportional to its external pressure. Since lower pressures are needed to reach a given coverage on stronger binding sites, adsorption rates are correspondingly smaller. Very recent experimental results of adsorption kinetics on closed nanotube bundles show this same trend for various adsorbates, providing basic support to our theoretical approach.7,8 Despite this initial agreement between the theoretical and experimental results, the limitations of our initial model preclude a meaningful direct comparison for some systems. Our theoretical simulations used a one-dimensional lattice to represent the adsorption sites along a groove between two nanotubes or a single strip on the outside of the tube. Actually, if we were to
10.1021/jp905562y CCC: $40.75 2009 American Chemical Society Published on Web 09/09/2009
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Burde et al. Applying the Kinetic Monte Carlo Algorithm to the Lattice-Gas Model
Figure 1. Modeling the external surface of a nanotube bundle as a heterogeneous lattice of sites. The solid lines in the graph at the top represent equipotentials (in K) for a CF4 molecule near the external surface of a bundle: The shaded areas represent portions of the cross section of two carbon nanotubes, and the dashed circles show the approximate positions of particles adsorbed in this situation. The y-axis is along the axis of the tubes. The picture at the bottom represents the corresponding lattice assigned to the groove (vertical chain of sites in the middle) and edge sites (vertical chains to the left and right).
represent the kinetics of adsorption on the whole external surface of a bundle, we have to consider the fact that the monolayer completion involves the successive adsorption of several stripes of atoms as the external pressure is raised, beginning at the groove sites and continuing on weaker adsorbing sites. For a given bundle of tubes, the monolayer includes a different number of stripes depending on the size of the adsorbate. Simulations and experiments show for example that a large molecule like CF4 would form a three-stripe monolayer,14 while a monolayer comprised of five lines is expected for smaller adsorbates like CH4, Xe, or Ar.15–19 The top panel in Figure 1 shows the equipotential lines corresponding to the external potential that a CF4 molecule would feel in the vicinity of the external surface of a bundle.14 The dashed circles show the approximate positions of the adsorbed particles, while the shaded areas represent the nanotubes. Note that the groove has a higher binding energy than the outer lines because of the addition of the contributions from two nearby nanotubes. In this work, we investigate the effect of having sites with different binding energies on the overall adsorption kinetics of the system. To achieve this, we perform simulations where we consider the rate at which particles adsorb on two-dimensional, heterogeneous lattices formed by three and five chains of adsorption sites.
We used a lattice-gas model of adsorption similar to what we used in our previous study4 to represent the adsorption of gases on the nanotube bundle. In this case, we represent the adsorption sites on the external surface of the bundle with a two-dimensional lattice made out of three lines of sites. The central line, representing the groove, has a stronger binding energy due to the interaction of particles in it with the nanotubes on either side, while the outer lines (edges) have weaker binding energies mainly arising from the contribution of a single nanotube. In order to model infinitely long lines, we consider periodic boundary conditions along their direction (corresponding to the y-axis in Figure 1). Likewise, the three lines represent a unit cell on the external surface and therefore periodic boundary conditions are also considered in the perpendicular direction (corresponding to the x-axis in Figure 1). For the fiveline lattice, we consider an additional line next to each one of the edges. We assign binding energies εm to the sites in the groove line, ε1 to the sites along the lines adjacent to the groove (as shown in Figure 1), and ε2 to the sites on the outermost edges. The total energy of a particle adsorbed on a particular site is the sum of the binding energy corresponding to that site and the particle-particle interaction (J) contribution coming from its first neighbors. We use these energies to compute the probabilities of adsorption, desorption, and diffusion as we did in our previous simulation study.6 In the present case, we consider diffusion in the system, both up and down each individual chain as well as movements between lines, by allowing the particles to “jump” between adjacent sites. With the transition probabilities calculated for every possible change in state, the KMC algorithm is executed following the steps described before:6 (1) We start with an empty lattice exposed to a gas at a given temperature T and chemical potential µ. (2) The probabilities of transition (corresponding to adsorption, desorption, or diffusion processes) to every possible state are evaluated.6 (3) A single transition is picked according to a specific selection rule.6,13 (4) The time and corresponding coverage are updated and recorded. Steps 2-4 are repeated until equilibrium is achieved; i.e., the coverage does not change anymore, reaching the equilibrium value neq (T, µ) corresponding to the given temperature and chemical potential. We run different simulations for increasing values of the chemical potential to explore the kinetics that leads to a range of increasing coverages up to monolayer completion (corresponding to fractional coverage equal to one). Simulation Results I. Three-Line Lattice. Our first consideration was to characterize the time evolution of the entire heterogeneous lattice for the noninteracting case (J ) 0). To accomplish this, we begin by plotting the fractional coverage as a function of time for several values of the chemical potential, as shown in Figure 2. At first glance, these results seem very similar to those of the homogeneous case.6 As the coverage increases with time, the overall evolution (top panel) follows what appears to be an exponential path to equilibrium characterized by a final coverage neq given by
Kinetics of External Adsorption on Nanotube Bundles
neq(T, µ) )
1/3 2/3 + β(εm-µ) 1+e 1 + eβ(ε1-µ)
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(1)
where µ is the chemical potential and β is the inverse of the temperature T, β ) 1/kBT. Furthermore, the equilibration time of the system (time needed to reach the flat portion of the curve) decreases monotonically as the chemical potential increases and approaches zero for the highest coverages, precisely what we saw for a uniform lattice. Upon closer inspection, however, we find that the heterogeneity of the lattice affects the adsorption kinetics in several distinctive ways. The overall coverage on the lattice arises from the sum of several components, in this case the coverage on the groove and on the two adjacent lines. When these partial coverages are considered separately, as shown in the bottom panel of Figure 2, the deviation from the behavior found in the previous study is readily seen, especially in the contribution of the outer lines (edges) that clearly does not follow an exponential time dependence. Moreover, the adsorption rate on these weaker binding sites appears to be much slower than that on the groove sites, contrary to what we would expect based solely on the individual binding strength of each group of sites (see the discussion of our previous study in the Introduction).6 Considering the equilibration time as a function of coverage, we see a large effect because of the heterogeneity in adsorption energies. Figure 3 shows the equilibration time as a function of coverage for several temperatures. In the homogeneous case, we observed a linear relationship between the waiting time and coverage.6 This linearity is lost for the heterogeneous lattice, particularly as the temperature falls. In order to analyze the magnitude of this deviation, we compare in Figure 4 the equilibration time for the heterogeneous lattice, featuring two binding energies, to the equilibration times
Figure 3. Top panel: Equilibration time as a function of coverage for the heterogeneous lattice at a range of temperatures. The bottom panel illustrates the population of the lattice at equilibrium for different values of the reduced chemical potential βµ, at T ) 80 K.
Figure 4. The upper and lower curves represent the equilibration time versus coverage for a homogeneous lattice of strong and weak binding sites, respectively. The middle line shows the dependence of the waiting time of the heterogeneous system on the coverage (T ) 120 K).
Figure 2. Upper panel: From bottom to upper curves, fractional coverage as a function of time for increasing values of chemical potential (εm ) -400 K, ε1 ) -200 K, T ) 90 K). Lower panel: The overall fractional coverage as a function of time corresponding to a final coverage of 0.7 (upper curve) with its contributions from the edges (squares) and groove (circles).
of homogeneous lattices of sites with only the stronger or only the weaker energies. At high coverages, there is very little difference between the waiting times of the heterogeneous lattice and those of the weak binding homogeneous lattice, while significant deviations are seen at the lowest coverages. The results at high coverage can be expected on the basis of the fact that for such high values of the chemical potential the high binding sites on the groove get filled almost instantly and the kinetics is dominated by the weaker site rates. On the other hand, at low values of the chemical potential, when the heterogeneous lattice is filling only the groove sites (up to fractional coverages less than 0.2), a dramatic increase in the adsorption rates of these strong binding sites is observed when they form part of the heterogeneous lattice. For example, when the fractional coverage of the three-line lattice is 0.1 (roughly corresponding to a fractional coverage of 0.3 for the total of groove sites), the waiting time is reduced almost by half.
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Figure 6. Equilibration time as a function of the ratio ∆ε/kBT (∆ε ) ε1 - εm) for different values of the equilibrium fractional coverage.
Figure 5. Top panel: The origin of the particles occupying the groove; the majority of particles adsorbed in the groove originally adsorbed in the edges. Bottom panel: A similar breakdown of the coverage of the edges is shown; most particles adsorbed in the edges come directly from the gas.
When we compare the equilibration process for each component of the lattice to their behavior as they would perform the same operation independently at an intermediate overall coverage, we observe that the outer lines reach equilibrium more slowly when they are a part of a heterogeneous lattice, while the groove equilibrates much faster. This implies that, in the case of the heterogeneous lattice, diffusion across the sites is playing an important role in the kinetics of the system. Because the adsorption rate increases as the difference in energy decreases, it is faster for the system to fill first the outer lines temporarily and then pass the particles along to the groove. In this way, the outer line sites work as “precursor” states and diffusion across this gradient is much faster than direct adsorption from the gas to the groove. Indeed, we show in Figure 5 where the particles that inhabit each phase were originally adsorbed from the gas phase. The top panel in Figure 5 shows the adsorption in the groove where we observe that a vast majority of its particles were originally adsorbed on the edges and diffused into the groove. Conversely, the bottom panel shows the coverage in the edges, where most of the particles were directly adsorbed from the gas to these sites. In this case, only a few particles diffuse from the groove to the edges. In Figure 6, we consider the dependence of the waiting time on the quantity β∆ε, with ∆ε being the difference in binding energy between the weaker and stronger binding sites. Because of the nonlinear response of the system with coverage (see Figure 3), we must consider each coverage separately rather than looking at the slope of a straight line, as we did in our previous work for the homogeneous lattice. For each coverage, we observe an exponential dependence with β∆ε of the form a1 exp(a2β∆ε), with the coefficients a1 and a2 being functions of the coverage. In particular, we observe that the exponential dependence gets stronger (with a2 > 1) as the coverage decreases
Figure 7. Effect of interactions on the equiliation times (T ) 90 K).
and adsorption through diffusion from the edge sites plays the most important role. Figure 7 illustrates the effect of adsorbate-adsorbate interactions on the waiting times; this is similar to the homogeneous case where repulsive interactions cause an overall decrease in the waiting times (since the effective binding energy of the site is decreased) and attractive interactions produce the opposite effect. II. Five-Line Lattice. When we consider a five-line lattice by adding two more outer lines to the system of three, we observe that the adsorption kinetics can be explained on the basis of the results for the three-line system. Figure 8 shows the time evolution of the overall coverage and the individual contributions from each group of sites. Again, we observe a notorious slowing down on the final occupancy of the weakest sites (the outer edges) due to diffusion of particles from these outer sites to stronger sites; as a consequence, a nonexponential dependence develops for the time evolution of the coverage. In Figure 9, we analyze the origin of the particles that end up adsorbed in each group of sites. The top panel shows that the groove is mainly filled with particles that were originally adsorbed from the gas on the outer edges and diffuse toward the groove sites. Smaller contributions come from particles that were originally adsorbed on the inner edges, and only a few percent are directly adsorbed from the gas on the groove sites. Similarly, the inner edge sites (see middle panel) become occupied mostly from particles coming from the outer edges with smaller contributions from direct adsorption from the gas and from particles coming from the groove. Finally, adsorption on the outer edge sites is mainly provided by direct adsorption
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Figure 8. Upper panel: From bottom to upper curves, fractional coverage as a function of time for increasing values of chemical potential (T ) 90 K). Lower panel: The overall fractional coverage as a function of time corresponding to a final coverage of 0.7 (upper curve) with its contributions from the edges (squares) and groove (circles).
from the gas; the second largest portion of particles comes from inner edges, while only a small contribution to the outer edge coverage comes from particles initially adsorbed on the groove sites (bottom panel). Discussion and Conclusions To investigate the effects of surface heterogeneity on the adsorption kinetics on the outside of a nanotube bundle, we have modeled the external surface as a strong binding line of sites bordered on either side by weaker binding lines. Particle uptake on each line of sites involves not only processes of direct adsorption from the gas phase but also the migration of particles between neighboring lines. The largest effect occurs during the occupancy of the strongest binding sites; the groove receives relatively few particles from the gas and instead sees the outer lines as a sort of infinite reservoir from which it can adsorb particles more quickly due to the smaller energy gap. The diffusion of particles from the outer lines results to be the primary mechanism for filling the groove; in this way, the system is able to speed up the adsorption on the groove by using adsorption on the outer lines as intermediate states. The particle migration from the outer sites to the central ones is responsible for the deviations from the exponential time evolution seen previously for a homogeneous lattice. Even at high coverages, when particles will end up filling both kinds of sites, this is still the dominant process to occupy the groove sites. This flow of particles slows down the filling of the weaker sites so much that the adsorption rate on the groove is greater than that on the edges, just the opposite to what is expected on the basis of only the binding energy values on each line. The overall equilibration time as a function of coverage of the heterogeneous system is greater than that of a uniform lattice
Figure 9. Top panel: The origin of the particles occupying the groove; the majority of particles adsorbed in the groove originally adsorbed in the outer and inner edges. A similar breakdown for the coverage of the edges is shown in the two bottom panels.
of weak binding sites but smaller than that of a lattice of strong binding sites. In particular, while the equilibration time is only slightly greater than the one of weaker uniform lattice at high coverages, the equilibration is considerably accelerated at low coverages. This difference causes the deviation from the linear dependence with coverage seen previously for homogeneous systems, especially as the temperature is lowered. As a consequence, the dependence of the equilibration times on β∆ε is also a function of the coverage. The results for the five-line system show that similar mechanisms occur during the adsorption on each line: strong binding sites are mostly populated by particles diffusing from neighboring weaker binding lines. Moreover, these results allow us to arrive at a general principle that rules the kinetic behavior in the presence of surface heterogeneity: since adsorption is faster for weaker binding sites, adsorption on stronger sites occurs mostly through diffusion from particles first adsorbed there rather than from direct adsorption from the gas phase, producing the set of effects on the equilibration time mentioned before. With this principle in mind, it is possible to predict the kinetic processes for more complicated heterogeneous surfaces that involve, for example, seven or more lines. Some of the kinetic behavior trends found in this work have also been observed experimentally.7,8 In ref 7, the equilibration
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time as a function of fractional coverage is reported for Ar and CH4 as they are adsorbed on as-prepared (closed-ended) nanotube bundles at 77 K. In ref 8, similar measurements are presented for H2 at 40 K and for CF4 at three different temperatures (117, 122, and 128 K). In all cases, the observed decreasing trend with coverage is consistent with our simulation results for the external surface of a bundle. When we compare the Ar and CH4 data at 77 K, we observe that the equilibration times are longer for CH4 in the whole coverage range. This also agrees with our simulation results, since the energy gap is larger for CH4 (Figure 6). A similar situation occurs when we compare equilibration times for H2 and CF4 where we observe much longer equilibration times for the stronger bound CF4. The temperature dependence of CF4 equilibration time agrees with the overall tendency observed in the simulations that predict longer times for lower temperatures. Since the coverage does not increase exponentially with time due to transfer of particles between sites of different binding energy, it is also expected that particle desorption during temperature programmed desorption experiments will not follow the Polany-Wigner equation. For this reason, simulation results will be of particular value in this case and will be the subject of a future publication. Acknowledgment. We acknowledge the support provided for this study by the National Science Foundation through Grants DMR-0705077 and CBET-0746029. M.C. and C.L.P. also acknowledge the support received by the National Science Foundation REU program through the grant DMR-0552800. References and Notes (1) Thess, A.; Lee, R.; Nikolaev, P.; Dai, H.; Petit, P.; Robert, J.; Xu, C.; Lee, H. Y.; Kim, S. G.; Rinzler, A. G.; Colbert, D. T.; Scuseria, G. E.; Tomzak, D.; Fischer, J. E.; Smalley, R. E. Science 1996, 273, 483.
Burde et al. (2) Gatica, S. M.; Calbi, M. M.; Diehl, R. D.; Cole, M. W. J. Low Temp. Phys. 2008, 152, 89. (3) Migone, A. D.; Talapatra, S. In Encyclopedia of Nanoscience and Nanotechnology; Nalwa, H. S., Ed.; American Scientific Publishers: Los Angeles, CA, 2004; pp 749-767. (4) Johnson, J. K.; Cole, M. W. In Adsorption by Carbons; Bottani, E. J., Tascon J. M. D., Eds.; Elsevier Science Publishing: Amsterdam, The Netherlands, 2008; Chapter 9. Also, M. Mercedes Calbi, M. W. Cole, S. M. Gatica, M. J. Bojan, J. K. Johnson, Chapter 15, and A. D. Migone, Chapter 16, in the above referenced book. (5) Calbi, M. M.; Riccardo, J. L. Phys. ReV. Lett. 2005, 94, 246103. (6) Burde, J. T.; Calbi, M. M. J. Phys. Chem. C 2007, 111, 5057. (7) Rawat, D. S.; Calbi, M. M.; Migone, A. D. J. Phys. Chem. C 2007, 111, 12980. (8) Rawat, D. S.; Krungleviciute, V.; Heroux, L.; Bulut, M.; Calbi, M. M.; Migone, A. D. Langmuir 2008, 24, 13465. (9) See, for example: Ulbritch, H.; Moos, G.; Hertel, T. Phys. ReV. Lett. 2003, 90, 095501. Kuznetsova, A.; Yates, J. T., Jr.; Smalley, R. E. J. Chem. Phys. 2000, 112, 9590. A brief (not comprehensive) review of literature involving this method was presented in ref 8. (10) See: Kondratyuk, P.; Wang, Y.; Liu, J.; Johnson, J. K.; Yates, J. T., Jr. J. Phys. Chem. C 2007, 111, 4578, and references there in. (11) See, for example, the discussion in: Calbi, M. M.; Toigo, F.; Cole, M. W. http://xxx.lanl.gov/abs/cond-mat/0406521. (12) Yang, R. T. Gas Separation by Adsorption Processes; Butterworths Publishers: Stoneham, MA, 1987. (13) Bulnes, F. M.; Pereyra, V. D.; Riccardo, J. L. Phys. ReV. E 1998, 58, 86. (14) Heroux, L.; Krungleviciute, V.; Calbi, M. M.; Migone, A. D. J. Chem. Phys. B 2006, 110, 12597. (15) Muris, M.; Dufau, N.; Bienfait, M.; Dupont-Pavlovsky, N.; Grillet, Y.; Palmari, J. P. Langmuir 2000, 16, 7019. (16) Bienfait, M.; Zeppenfeld, P.; Dupont-Pavlovsky, N.; Muris, M.; Johnson, M. R.; Wilson, T.; DePies, M.; Vilches, O. E. Phys. ReV. B 2004, 70, 035410. (17) Rols, S.; Johnson, M. R.; Zeppenfeld, P.; Bienfait, M.; Vilches, O. E.; Schneble, J. Phys. ReV. B 2005, 71, 155411. (18) Calbi, M. M.; Cole, M. W. Phys. ReV. B 2002, 66, 115413. (19) Gatica, S. M.; Bojan, M. J.; Stan, G.; Cole, M. W. J. Chem. Phys. 2001, 114, 3765.
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