J. Phys. Chem. C 2007, 111, 5057-5063
5057
Physisorption Kinetics in Carbon Nanotube Bundles Jared T. Burde and M. Mercedes Calbi* Department of Physics, Southern Illinois UniVersity, Carbondale, Illinois 62901-4401 ReceiVed: August 22, 2006; In Final Form: December 21, 2006
The kinetics of gas uptake on different regions of carbon nanotube bundles is investigated by means of a kinetic Monte Carlo scheme. A lattice-gas description is used to model the adsorption of particles on a onedimensional chain of sites under two types of dynamics: (a) external kinetics, in which the chain is on the bundle’s external surface directly exposed to the gas, and (b) pore-like kinetics, expected to occur inside the tubes and interstitial channels, where adsorption occurs via gas diffusion from the ends. From the time evolution of the coverage at a fixed temperature, equilibration times are obtained as a function of chemical potential (or amount adsorbed). The equilibration time of the external phase decreases linearly as the coverage increases toward monolayer completion; the rate at which this occurs strongly depends on the ratio between the binding energy and the temperature. Because of this dependence, unexpectedly long waiting times can be observed for very low coverages in systems with relatively high binding energies. The adsorption rate in pore-like phases is typically 2 orders of magnitude slower than that of external phases. We show how this large disparity between adsorption rates can hinder the observation of adsorption inside the tubes and in the interstitial channels during measurements of adsorption isotherms.
1. Introduction The unique geometrical and structural characteristics of carbon nanotubes have motivated extensive investigations of their performance as gas adsorbents.1,2 Many of these research efforts have been related to their capacity to generate onedimensional (or quasi-one-dimensional) phases of matter when various gases are adsorbed on their surfaces.1-32 When singlewalled carbon nanotubes are produced, they usually form ropes or bundles.33 These bundles consist of groups of nearly parallel nanotubes arranged in a triangular lattice, where the lattice constant is determined by tube-tube interactions. The tubes are generally closed by semi-spherical carbon caps that are spontaneously formed at their ends; opening the tubes can be achieved by various chemical or mechanical treatments. In addition to the adsorption sites present for a single tube (its internal and external surfaces), the bundles exhibit two other distinctive adsorption sites: the grooves formed on the exterior of the bundle where two adjacent tubes meet, and the interstitial channels (ICs) that lie in between three tubes in the interior of the bundle.17 Depending on the adsorbate and the size of the tubes, gases have been found to form a variety of phases when adsorbed on these bundles, including dimensional crossover behavior from one-dimensional to two- , three-, and even higher dimensional phases.20-23 In general, every site is characterized by a specific value of the binding energy, which determines the shape of the adsorption isotherms and the coverage dependence of the isosteric heat of adsorption. Although most studies agree on the adsorption characteristics of the external sites, the interpretation of the isotherms based on the occupation of internal sites is more ambiguous and somewhat controversial.5,13-15,34 This may be due in part to the fact that the external sites (grooves and external surface) are exposed directly to the gas phase during equilibration while the * Corresponding author. E-mail:
[email protected]. Phone: 618453-2048. Fax: 618-453-1056.
internal sites (ICs and inside the tubes) are occupied as a result of gas diffusion through the ends of the tubes. Much longer equilibration times are required in the latter case to allow this internal adsorption to happen. The presence of energy barriers at the openings of the pores may slow down the process even further, as we have shown recently in the case of H2 absorbing in the interstitial channels between tubes of radii around 7 Å.35 At present, there is little information on the adsorption rates corresponding to the various sites in a bundle. This work aims at exploring those rates as a function of temperature and amount of gas that is adsorbed. Because adsorption rates are expected to differ between binding sites, knowledge of the relationship between these rates can provide another method for identifying these sites. Characteristic times of adsorption/desorption processes are also particularly important when considering gas separation applications. Kinetic separation can be achieved when different molecules exhibit different diffusion and/or adsorption rates. In such a case, the separation mechanism is driven by controlling the time of exposure. For example, a particular time of exposure can be chosen so that when a binary mixture of gases interacts with an adsorbent the species with the longest equilibration time is not significantly adsorbed and can be mostly recovered from the mixture. Because of this, obtaining information on adsorption rates for different gases is the first step of assessing the separation performance of adsorbents through kinetically selective processes. Although these kinetic properties have been investigated in other porous carbon materials,36-39 little information is currently available for carbon nanotube bundles despite the fact that they are very likely to be useful in such applications. Because of the large aspect ratio of the tubes, molecules adsorbed in nanotube bundles assemble in one-dimensional chains (or lines) that eventually lead to the formation of oneto three-dimensional adsorbed phases. Depending on where each line forms, its adsorption sites exhibit different binding energies
10.1021/jp065428k CCC: $37.00 © 2007 American Chemical Society Published on Web 03/14/2007
5058 J. Phys. Chem. C, Vol. 111, No. 13, 2007
Burde and Calbi
and kinetic restrictions. Typically, these lines are present in three different regions of the bundle: (1) The bundle’s external surfaces: Depending on the external pressure of the gas and the temperature, one or more lines are adsorbed, starting with the formation of a one-dimensional phase in the grooves at relatively low temperatures and pressures.18,19 (2) The tube’s interior: Adsorbed phases in this region can consist of a single line along the axis of the tube and/or several lines formed near the inner wall of the tube, depending on the relative size of the adsorbate compared to the diameter of the tube and also on the external pressure.25-29 (3) Interstitial spaces: In most cases, for typical sizes of nanotubes and adsorbates, only a single line can be formed in this region. At times, however, wider ICs can hold additional lines, especially if the packing of the bundle has defects.30-32,34 In this work, we focus on modeling the adsorption kinetics of molecules on a one-dimensional chain of sites under different thermodynamic and kinetic conditions. This single line of sites represents the adsorbing surface when only one-dimensional phases are formed; it is also the elemental structure for modeling and analyzing the adsorption kinetics of higher dimensional phases. To understand the variety of kinetic behaviors that may be present during gas physisorption on a bundle, we consider two types of dynamics: (a) adsorption kinetics of an external line (in the exterior of the bundle), where the surface is directly exposed to the external gas, and (b) adsorption kinetics of a pore-like phase (in the tube’s interior and ICs), where we assume that gas adsorption occurs inside the pore via gas diffusion from the ends. To model this difference, we consider that, in the groove case, adsorption/desorption processes, as well as displacements (or jumps), happen along the whole chain. In the case of pores, we limit adsorption/desorption processes to the end sites; occupation of internal sites can occur only as a consequence of particle displacements along the chain. From the time evolution of the gas uptake, we can derive the equilibration time as a function of the equilibrium coverage for different temperatures. During isotherm measurements, adsorption kinetics is usually monitored by waiting a sufficient amount of time to ensure that the adsorbed phase has reached equilibrium with the external gas. Those waiting (or equilibration) times, which are related to the inverse of the adsorption rates, can be directly compared with our theoretical results. In fact, it was the observation, during adsorption measurements in nanotube bundles, that waiting times monotonically decreased as the coverage increased from very low values to the monolayer coverage13,41,42 that helped prompt this work. In particular, in recent measurements of CF4 adsorption at very low coverages, the values of the waiting times were observed to be unexpectedly high, assuming that adsorption was primarily taking place on the exterior of the bundle. This work provides a first explanation of how that can be possible. 2. Kinetic Monte Carlo Scheme for Adsorption and Diffusion We consider a set of particles that can be adsorbed on a onedimensional array of Ns sites. We adopt a lattice gas model where the energy Ei of a particle on site i is given by
Ei ) i + J
∑ sj
(1)
j,NN
where i is the adsorption energy and the second term represents the interaction energy (of strength J) between the particle on site i and its nearest neighbors (NN). The variable sj takes the value 0 if the site is empty or 1 if it is occupied.
To investigate the time evolution of the coverage n(t) (ratio between number of adsorbed particles and Ns) as the system approaches the equilibrium state, we implement the kinetic Monte Carlo scheme used in our previous work.35,40 This evolution is driven by transition probabilities Wij corresponding to transitions from an initial state i to a final state j. In the present work, we consider adsorption/desorption processes as well as displacements (or jumps) along the chain. The transition probabilities are chosen so that the detailed balance principle is satisfied; in this way, the eventual achievement of proper thermodynamic equilibrium is guaranteed. For particle jumps, we calculated the probabilities as
Wij ) exp[-β(Ej - Ei )] Wji
(2)
where β ) 1/kBT (T is the temperature of the system and kB is the Boltzman constant). The probabilities for adsorption and desorption on site i are determined as
Wads ) exp[-β(Ei - µ)] Wdes
(3)
where µ is the chemical potential of the external gas interacting with the adsorbent lattice. Starting from an initial state with coverage n0 and fixing the values of T and µ, the simulation is based on the following steps: 1. For all of the possible final states j from the initial state i, the transition probabilities are calculated. The system makes the transition to the kth state that satisfies the condition
1 W
k-1
∑ j)1
Wij < R1 e
1
k
∑ Wij
W j)1
(4)
where R1 is a random number between 0 and 1 and
W)
Wij ∑ i, j
2. After the transition is made, the time variable t is incremented by the interval ∆t
∆t)-
1 ln R2 W
(5)
where R2 is another random number between 0 and 1. Steps 1 and 2 are repeated until the equilibrium coverage for the given values of T and µ is reached. 3. Adsorption Kinetics of External Phases We start the analysis of adsorption kinetics in the simplest case possible, that is, adsorption on a homogeneous chain of sites with adsorption energy (where < 0) in which interactions between particles are neglected. This simple model provides a very important reference from which deviations in real systems (either simulated or experimental) can be better analyzed. In addition, the simulation results for this case can be understood and tested using an analytical model as shown below. 3.1. Simulation Results. Figure 1 shows the coverage as a function of time at a fixed value of the temperature for different values of the chemical potential. Because the observed behavior only depends on the ratio β and the difference x ) β( - µ), the results are given in terms of those variables. For a specific system at a given temperature, the ratio β is fixed and x
Physisorption Kinetics in Carbon Nanotube Bundles
J. Phys. Chem. C, Vol. 111, No. 13, 2007 5059 adsorption kinetics in which the time dependence of the coverage is determined by the rate equation
dn ) Wads(1 - n) - Wdes n dt
(8)
In this equation, every adsorbed molecule has a probability Wdes of being desorbed, every empty site may be occupied with probability Wads, and these processes happen independently for every site of the lattice. By defining a new dimensionless time variable t* as t* ) tWdes and using eq 3 the rate equation becomes
dn ) e-β(-µ)(1 - n) - n dt* Figure 1. Time evolution of fractional coverage n ) Nads/Ns for a homogeneous chain of sites at fixed temperature (β ) -1.4) for different values of the chemical potential given by x ) β( - µ).
Figure 2. Equilibration time τ as a function of the equilibrium coverage for different values of the temperature. The inset shows the temperature dependence of the slope of the resulting lines.
indicates the value of the reduced chemical potential βµ. Every curve follows an evolution given by
n(t) ) neq[1 - e-t/τ] + n0e-t/τ
(6)
where n0 is the initial coverage, and neq is the equilibrium value of the coverage given as
neq(T,µ) )
1 1 + eβ(-µ)
(7)
due to the independence of the adsorption sites. The characteristic time τ (or the adsorption rate 1/τ) provides an indication of the time needed to reach equilibrium, which clearly decreases as the chemical potential (and consequently neq) increases. Figure 2 shows the characteristic time τ as a function of the equilibrium coverage for different values of the ratio β. We observe that τ decreases linearly as the coverage increases, with a temperaturedependent slope that increases with the magnitude of β. Moreover, the slope follows the exponential dependence shown in the inset of Figure 2. This kind of universal behavior means, for example, that, for a fixed value of the temperature and equilibrium coverage, systems with greater binding energies will exhibit longer equilibration times. 3.2. Analytical Calculations. The simulation results presented above can be understood considering a simple model of
(9)
The solution of this equation is given by eq 6 with the characteristic time
τ* ) e-β(1 - neq)
(10)
as found in the simulation results. This type of time evolution corresponds to the Langmuir kinetics of adsorption in which the adsorption rate is proportional to the external gas pressure (or eβµ) and to the empty space of the lattice (1 - n). In this context, for high binding energies or low temperatures, smaller values of pressure (and therefore longer times) are required to achieve a given equilibrium coverage. This effect can be rather dramatic because of the exponential dependence of the waiting time on β shown in eq 10 For example, because the binding energy of methane in the grooves is around 350 K larger than that of Ar, waiting times for methane at 150 K are expected to be 10 times longer. This result also has interesting implications for the adsorption kinetics of a binary mixture of gases with different binding energies. On the basis of the previous results, the following phenomenon is expected to happen: At partial pressures such that either species is expected to cover, for example, 75% of the lattice if it were the only one present, there will be a competition based on both the binding energies and the adsorption rates. Because the species with the smaller binding energy will adsorb faster, it is expected that, before reaching equilibrium, this species will be the favored one, contrary to what will eventually happen in equilibrium. Therefore, the weaker binding species is expected, initially, to reach coverages larger than the equilibrium value. This kind of phenomenon offers another type of kinetic selective process that could be exploited for separation processes. 3.3. Effects of Interparticle Interactions. Interactions between adsorbed particles modify these findings because the sites cannot be considered as independent. This is especially true if gas-gas interactions are comparable to the gas-surface interaction energy (as happens in the case of weakly binding substrates). In Figure 3, we illustrate the effect of attractive and repulsive interactions on the time evolution of the coverage. Although the exponential behavior of the coverage is still observed, the equilibrium coverage is greater in the attractive case and smaller when repulsive interactions are present. This follows directly from eqs 1 and 7 because attractive interactions have the effect of inducing additional adsorption for the same value of chemical potential by increasing the “effective” binding energy. Alternatively, repulsive interactions tend to impede adsorption so the equilibrium coverage is reduced. The deviation of the equilibration times from the noninteracting case is accordingly shown in Figure 4. 4. Adsorption Kinetics of Pore-like Phases Adsorption in internal pores (or channels) requires particle transport from the openings to the inside of the pores. In this
5060 J. Phys. Chem. C, Vol. 111, No. 13, 2007
Figure 3. Effect of adsorbate-adsorbate interactions on the time evolution of the coverage. The top curve corresponds to attractive interactions, and the bottom curve results in the case on repulsive interactions. The middle curve corresponds to the noninteracting case.
Burde and Calbi
Figure 5. Potential energy of a C60 molecule along the axis of an open tube, near its opening. Positive values of the abscissa correspond to the interior of the tube. Each curve corresponds to a different value of the radius (R) of the tube. The value of σgc ) 6.4 Å was taken from ref 17.
Figure 4. Equilibration time as a function of equilibrium coverage for the cases illustrated in Figure 3.
case, only the end sites are in direct contact with the external gas. To analyze the effect of this restriction on the adsorption kinetics, we consider gas adsorption on a chain of sites in which the adsorption/desorption processes are limited to the end sites. Occupation of internal sites happens as a consequence of particle displacements along the chain. Generally speaking, the strongest binding inside a pore occurs when R ∼ 1.1σgc, where R is the pore’s radius and σgc is the length parameter for the adsorbate-carbon Lennard-Jones potential. Only a one-dimensional phase can be formed along the axis of such a pore. By decreasing the radius from that value, the binding energy is reduced inside the tube; an energy barrier develops at its opening when R ∼ 0.95σgc. Eventually, when R is smaller than ∼0.93σgc the pore becomes too small to adsorb the gas. In Figure 5, we illustrate the evolution of the adsorbing potential for C60 as the radius of the tube changes. The potential was calculated following the standard procedure given in ref 28. When R ∼ 0.95σgc, there is an adsorption site at the entrance of the tube; a large barrier must be overcome to reach the interior of the tube, where the binding energy is approximately the same as that of the external site. For a given tube radius, the effective radius of the ICs in a perfect bundle is much smaller than that of the tubes but the development of the energy barrier as the size of the channel is changed is approximately the same. Figure 6 shows the typical values of the radius of the tube for which an energy barrier at the opening of the tube (R ) 0.95σgc) or IC (RIC ) 0.95σgc) is expected, depending on the gas considered. For example, for a perfect bundle of tubes with
Figure 6. Typical values of the radius of the tube for which an energy barrier at the entrance of the pore (tube or IC) is expected for different gases (according to the condition Rpore ) 0.95σgc; see Figure 5). The molecules are assumed to be spherical; representative values of σgc were extracted from ref 17.
R ) 7 Å an energy barrier is expected for H2 at the entrance of the ICs but no barrier is expected at the opening of the tubes for any of the adsorbates shown in the figure. It is important to mention that Figure 6 intends to show only representative values of the radius of the tubes for which an energy barrier may be present. The details of the adsorption behavior and the exact range of values of the radii must be determined by studying each specific system separately. According to the discussion above, we consider the following two cases in the simulations: a homogeneous chain, where all of the sites are characterized by the same binding energy, and a chain where the end sites have larger binding energies than the internal sites, simulating the effect of a diffusive barrier at the ends. In Figure 7, we show the equilibration times in the pore-like homogeneous chain for several values of β. Although a decreasing trend similar to that of external systems is obtained, equilibration times are typically 1-2 orders of magnitude longer, which means that the adsorption kinetics is considerably slower. The slope of the lines exhibits the same exponential dependence on β as seen before; however, the equilibration time for monolayer completion is generally different from zero; rather, it decreases exponentially with β.
Physisorption Kinetics in Carbon Nanotube Bundles
Figure 7. Equilibration time as a function of the equilibrium coverage for pore-like kinetics. The temperature dependence of the slope of the lines and of the equilibration time at monolayer completion (τm) is shown in the insets.
Figure 8. Effect of energy barriers on the equilibration time for porelike phases. The value of β ) -2.19 and the height of the barrier is βb ) -1.
We performed the simulations for a fixed value of the length of the tube (or number of sites); because adsorption is allowed to happen only at the end sites, the equilibration time is also expected to depend on the length of the tube considered. Simulations performed for different lengths of the tube showed that the equilibration time increases linearly with the tube’s length. However, the ratio between equilibration times corresponding to different values of β, shown in Figure 7, keeps the same value regardless of the tube length. Results from Figure 8 illustrate how energy barriers at the ends of the chain can further slow the kinetics. Although the coverage dependence follows a similar decreasing trend to that seen in the “no barrier” case, a change in the shape of the curve is observed. The largest difference occurs at half filling where the diffusion process, affected by the presence of the barrier, plays its largest role in determining the population of the internal sites. 5. Adsorption Kinetics in Systems with Pores and Grooves When gas is adsorbed in different regions of a nanotube bundle, a combination of both cases discussed above is present. To show how different binding energies and kinetic restrictions in these elemental lines would affect the overall kinetic behavior of the adsorbed phase when they are part of a higher dimensional system, we consider a system that contains three chains of
J. Phys. Chem. C, Vol. 111, No. 13, 2007 5061
Figure 9. Time evolution of uptake for a system composed by two external chains and one pore-like chain. Panel b displays the complete time evolution, and panel a shows only the first part of the evolution process, when external phases have reached equilibrium but the porelike uptake is still far away from that state.
sites: two external and one pore-like, each one characterized by a different binding energy. The two external chains model the adsorption on an external groove (with binding energy gr) and on the external surface of the tubes (ext), while the porelike chain simulates adsorption inside a tube (tube). The binding energies were chosen so that they represent the adsorption scenario of relatively large adsorbates, gr > tube > ext. In particular, we consider tube ) 0.88 gr and ext ) 0.75 gr . Even though this is a simple model of adsorption in a bundle, this system contains all of the basic ingredients that are expected to play a role in determining the overall adsorption kinetics in a bundle. In Figure 9, we illustrate the typical time evolution of the coverage. The top curve corresponds to the total coverage of the system while the others represent the partial coverage of each of the three chains considered. One can clearly observe that, even though the final partial coverage is comparable for the three chains (something that is expected considering that binding energies are very close to each other), the external lines reach equilibrium in a much shorter period of time. This is also evident in the kink that the total coverage exhibits at an early time in the equilibration process (see the arrow in panel b). It is this large difference in rates that may create the false appearance of equilibrium at those early times, when the external chains have reached their equilibrium coverage but the porelike phase is still far from it (see left panel of Figure 9). A situation like that in Figure 9a shows how a system could (erroneously) appear to have reached equilibrium long before the system has actually reached that state. If such an experiment were cut short at that time, then significant internal adsorption, shown in panel b, would be undercounted. It is not possible to extract a single value of the equilibration time from these curves because the coverage evolution for the external and pore-like phases are characterized by two time scales that are very different. A priori, one can expect that the experimental equilibration time will be dominated by the porelike behavior; however, the coverage dependence in this case will also be determined by the values of the binding energies corresponding to each line. For example, if the binding energy in the pore-like phase is smaller than that in the groove-like phases, then no appreciable adsorption will occur at the lowest coverages and waiting times would be short. As the total coverage increases, the pore-like phase will start to form, so longer waiting times would be expected. Eventually, the waiting times would decrease again as all of the sites are populated.
5062 J. Phys. Chem. C, Vol. 111, No. 13, 2007 6. Discussion and Conclusions We have investigated the kinetics of adsorption of particles on one-dimensional chains of sites, considering two main types of dynamics to simulate kinetics on both external and internal sites of a nanotube bundle. In both cases, equilibration times are observed to decrease as more particles are adsorbed at a given temperature. This is in agreement with the reported trend in waiting times observed during adsorption measurements in both open-13 and closed-ended nanotube samples.41,42 For a nanotube bundle’s exterior, we expect to observe the kind of behavior shown in Figure 2 during the formation of the groove phase (one line). In that case, we have shown that the rate at which the waiting time decreases with coverage depends exponentially on the ratio of the binding energy to the temperature; therefore, unexpectedly large values of equilibration times can be observed (especially at low pressure) for high binding energies or low temperatures. Equilibration times as long as 22 h have been observed experimentally for CF4 adsorption at the lowest coverages (about 10% of a monolayer) at temperatures around 125 K.41,42 At first glance, these waiting times may seem unusually long for adsorption on external surfaces; however, it is possible to understand these results in terms of the large binding energy of CF4 on the grooves. Although our results are given in terms of reduced quantities (τ* ) τWdes), we can use eq 10, with ) 2450 K, T ) 125 K, and a waiting time of 22 h to obtain Wdes ∼ 1.3 × 1012 s-1, which is a typical value for physisorbed systems.43 In addition, preliminary experimental results of adsorption kinetics of CF4, Ar, CH4, and H2 on nanotube bundles show the same decreasing trend with coverage described above.41,42 A strict comparison with these results requires the consideration of adsorption on additional chains of sites on the external surface of the bundle, each one characterized by a different binding energy. Details of the agreement found for these systems are the subject of our current investigations and will be published elsewhere. When particles are directly adsorbed from the gas phase only at the ends and must diffuse from there to reach the internal spaces, equilibration times are typically 2 orders of magnitude longer. We have shown that this large difference may obscure the evidence of the presence of internal occupation while measuring adsorption isotherms in a bundle. In particular, these results show that adsorption kinetics needs to be monitored very carefully when internal sites are expected to be available; failure to allow those phases sufficient time to reach equilibrium may lead to incorrect conclusions concerning, for example, the percentage of open tubes present in the sample. As outlined in section 5, adsorption in those sites can be better revealed from the coverage dependence of the equilibration times, making adsorption kinetics a very useful way to determine the occupancy of different sites in a bundle. Our results are focused on the relatiVe magnitude of adsorption rates in different situations; therefore, adsorption rates are given in reduced dimensions. Obtaining their values in real time units requires additional information about pre-exponential factors for the transition probabilities Wij that are not yet available. Alternatively, by comparing our results with experimental measurements of waiting times it will be possible to obtain the first estimates of these parameters as we have done above for the case of CF4. This work intends to be a first step toward understanding adsorption kinetics in nanotube bundles. The implemented kinetic Monte Carlo scheme has proven to be a very useful method of providing general trends of the expected kinetic behavior in different conditions. In that respect, these results
Burde and Calbi are meant to supply valuable insight into experiments and stimulate more research work in an area of significant importance from both fundamental and practical perspectives. Acknowledgment. Acknowledgment is made to the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research. References and Notes (1) Migone, A. D.; Talapatra, S. In Encyclopedia of Nanoscience and Nanotechnology; Nalwa, S.; Ed.; American Scientific Publishers: Los Angeles, CA, 2004; Vol. 4, pp 749-767. (2) Adsorption by Carbons; Tascon, J. M. D., Ed.; Elsevier Science, in press, 2006. (3) Calbi, M. M.; Cole, M. W.; Gatica, S. M.; Bojan, M. J.; Stan, G. ReV. Mod. Phys. 2000, 73, 857. (4) Kuznetsova, A.; Yates, J. T., Jr.; Liu, J.; Smalley, R. E. J. Chem. Phys. 2000, 112, 9590. (5) Talapatra, S.; Migone, A. D. Phys. ReV. Lett. 2001, 87, 206106. (6) Talapatra, S.; Rawat, D. S.; Migone, A. D. J. Nanosci. Nanotechnol. 2002, 2, 467. (7) Talapatra, S.; Krungleviciute, V.; Migone, A. D. Phys. ReV. Lett. 2002, 89, 246106. (8) Krungleviciute, V.; Heroux, L.; Talapatra, S.; Migone, A. D. Nano Lett. 2004, 4, 1133. (9) 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. (10) Wilson, T.; Vilches, O. E. Low Temp. Phys. 2003, 29, 732. (11) Pearce, J. V.; Adams, M. A.; Vilches, O. E.; Johnson, M. R.; Glyde, H. R. Phys. ReV. Lett. 2005, 95, 185302. (12) Antsygina, T. N.; Poltavsky, I. I.; Chishko, K. A.; Wilson, T. A.; Vilches, O. E. Low Temp. Phys. 2005, 31, 1007. (13) Rols, S.; Johnson, M. R.; Zeppenfeld, P.; Bienfait, M.; Vilches, O. E.; Schneble, J. Phys. ReV. B 2005, 71, 155411. (14) Jakubek, Z. J.; Simard, B. Langmuir 2005, 21, 10730. (15) Jakubek, Z. J.; Simard, B. Langmuir 2004, 20, 5940. (16) Teizer, W.; Hallock, R. B.; Dujardin, E.; Ebbesen, T. W. Phys. ReV. Lett. 1999, 82, 5305. (17) Stan, G.; Bojan, M. J.; Curtarolo, S.; Gatica, S.; Cole, M. W. Phys. ReV. B 2000, 62, 2173. (18) Gatica, S. M.; Bojan, M. J.; Stan, G.; Cole, M. W. J. Chem. Phys. 2001, 114, 3765. (19) Calbi, M. M.; Gatica, S. M.; Bojan, M. J.; Cole, M. W. J. Chem. Phys. 2001, 115, 9975. (20) Calbi, M. M.; Cole, M. W. Phys. ReV. B 2002, 66, 115413. (21) Kostov, M. K.; Calbi, M. M.; Cole, M. W. Phys. ReV. B 2003, 68, 245403. (22) Ancilotto, F.; Gatica, S. M.; Cole, M. W. J. Low Temp. Phys. 2005, 138, 201. (23) Ancilotto, F.; Calbi, M. M.; Gatica, S. M.; Cole, M. W. Phys. ReV. B 2004, 70, 165422. (24) Heroux, L.; Krungleviciute, V.; Calbi, M. M.; Migone, A. D. J. Phys. Chem. B 2006, 110, 12597. (25) Gatica, S. M.; Cole, M. W. Phys. ReV. E 2005, 72, 041602. (26) Trasca, R. A.; Calbi, M. M.; Cole, M. W. Phys. ReV. E 2002, 65, 061607. (27) Gatica, S. M.; Stan, G.; Calbi, M. M.; Johnson, J. K.; Cole, M. W. J. Low Temp. Phys. 2000, 120, 337. (28) Stan, G.; Cole, M. W. Surf. Sci. 1998, 395, 280. (29) Maddox, M. W.; Gubbins, K. E. Langmuir 1995, 11, 3988. (30) Calbi, M. M.; Toigo, F.; Cole, M. W. Phys. ReV. Lett. 2001, 86, 5062. (31) Calbi, M. M.; Toigo, F.; Cole, M. W. J. Low Temp. Phys. 2002, 126, 179. (32) Shi, W.; Johnson, J. K. Phys. ReV. Lett. 2003, 91, 015504. (33) Thess, A.; Lee, R.; Nikolaev, P.; Dai, H. J.; Petit, P.; Robert, J.; Xu, C. H.; Lee, Y. H.; Kim, S. G.; Rinzler, A. G.; Colbert, D. T.; Scuseria, G. E.; Tomanek, D.; Fischer, J. E.; Smalley, R. E. Science 1996, 273, 483. (34) Calbi, M. M.; Toigo, F.; Cole, M. W. http://xxx.lanl.gov/abs/condmat/0406521.
Physisorption Kinetics in Carbon Nanotube Bundles (35) Calbi, M. M.; Riccardo, J. L. Phys. ReV. Lett. 2005, 94, 246103. (36) Reid, C. R.; Thomas, K. M. Langmuir 1999, 15, 3206. (37) Reid, C. R.; O’koye, I. P.; Thomas, K. M. Langmuir 1998, 14, 2415. (38) Fletcher, A. J.; Yuzak, Y.; Thomas, K. M. Carbon 2006, 44, 989. (39) Zhao, X.; Villar-Rodil, S.; Fletcher, A. J.; Thomas, K. M. J. Phys. Chem. B 2006, 110, 9947.
J. Phys. Chem. C, Vol. 111, No. 13, 2007 5063 (40) 58, 86. (41) (42) 2005. (43) Verlag:
Bulnes, F. M.; Pereyra, V. D.; Riccardo, J. L. Phys. ReV. E 1998, Migone, A. D.; Rawat, D. Private communication. Heroux, L. M. S. Thesis Dissertation, Southern Illinois University, Kreuzer, H. J.; Gortel, Z. W. Physisorption Kinetics; SpringerBerlin Heidelberg, 1986.
